💡Okay, so I have this problem here: "If the line l passes through the origin and (3, -3), then its slope is ( )". The options are A: 135°, B: 45°, C: 45° or 135°, D: -45°. Hmm, I need to figure out the slope of the line. First, I remember that the slope of a line passing through two points (x1, y1) and (x2, y2) is calculated by the formula (y2 - y1)/(x2 - x1). In this case, the two points are the origin (0,0) and (3, -3). So, plugging these into the formula, I get:Slope (m) = (-3 - 0)/(3 - 0) = (-3)/3 = -1.Okay, so the slope is -1. Now, the question is asking for the angle, right? Because the options are in degrees. I think that relates to the angle of inclination of the line. The angle of inclination is the angle formed between the positive direction of the x-axis and the line, measured counterclockwise. I recall that the slope is equal to the tangent of that angle. So, if m = tan(θ), then θ = arctan(m). Since m is -1, θ = arctan(-1). Wait, arctan(-1) is... Hmm, I know that tan(45°) is 1, so tan(-45°) would be -1. But angles are typically measured as positive in the counterclockwise direction, so how does that work? If the slope is negative, that means the line is going downward from left to right. So, starting from the positive x-axis, if I go counterclockwise, I would reach 135°, which is in the second quadrant. Because tan(135°) is tan(180° - 45°) = -tan(45°) = -1. So, does that mean the angle is 135°? But wait, sometimes angles can be represented in different ways. For example, -45° is the same as 315°, but that's not one of the options. The options are 135°, 45°, 45° or 135°, and -45°. I think the angle of inclination is always between 0° and 180°, so 135° makes sense here. But why is -45° an option? Maybe because the slope is -1, which corresponds to an angle of -45°, but in the context of the angle of inclination, we take the positive angle, which is 135°. Wait, but sometimes people might refer to the angle as the smallest angle from the x-axis, which could be 45°, but since the slope is negative, it's actually 135°. So, I think 135° is the correct angle here. Let me double-check. If I have a slope of -1, the line goes down at a 45° angle from left to right. So, from the positive x-axis, going counterclockwise to the line would be 135°, and going clockwise would be -45°, but in terms of positive angles, 135° is the standard answer. So, looking back at the options: A is 135°, B is 45°, C is 45° or 135°, and D is -45°. I think A is the correct answer because it's the positive angle in the standard position. But wait, why is C an option? It says 45° or 135°. Is there a reason to think it could be either? Hmm, maybe if you consider the reference angle, which is 45°, but the actual angle of inclination is 135°. So, perhaps C is trying to trick you into thinking about the reference angle instead of the actual angle. But I think the question is specifically asking for the angle of inclination, which should be 135°. Alternatively, if the question was asking for the angle between the line and the x-axis, regardless of direction, it could be 45°, but since it's asking for the slope's angle, which is the angle of inclination, it should be 135°. Wait, but in some contexts, people might refer to the angle as the smallest angle, which would be 45°, but with the understanding that it's downward. But I think in mathematical terms, the angle of inclination is always between 0° and 180°, so 135° is correct. So, I think the answer is A: 135°. But I'm a bit confused because sometimes angles can be represented differently. Maybe I should visualize it. If I plot the points (0,0) and (3,-3), the line goes from the origin down to the right. So, from the positive x-axis, turning counterclockwise to reach the line would be 135°, which is in the second quadrant. That makes sense because the line is going downward, so it's in the second quadrant. If I think about the slope being -1, that's a steepness of 1, but in the negative direction. So, the angle is 135°, not 45°, because 45° would be a positive slope. Wait, but if I consider the angle between the line and the x-axis, it's 45°, but since it's going downward, it's 180° - 45° = 135°. So, yeah, 135° is the correct angle of inclination. Therefore, I think the correct answer is A: 135°.

💡Okay, so I've got this question about correlation coefficients. Let me try to figure it out step by step. First, the question says that we have a set of sample data points, each denoted as (x₁, y₁), (x₂, y₂), and so on up to (xₙ, yₙ). It mentions that n is at least 2, and importantly, the x-values aren't all the same. That probably means that there's some variation in the x-values, which is good because if all x-values were the same, the correlation coefficient might be undefined or something.Now, the key part is that all these sample points lie exactly on the line y = (1/2)x + 1. Hmm, so every single point (x_i, y_i) satisfies this equation. That means if I plug in any x_i into the equation, I'll get the corresponding y_i. I remember that the correlation coefficient measures how closely two variables are related in a linear way. It ranges from -1 to 1. If it's 1, it means a perfect positive linear relationship, and if it's -1, it's a perfect negative linear relationship. A 0 means no linear relationship.In this case, since all the points lie exactly on a straight line, that should mean the correlation is either 1 or -1, right? Because a perfect linear relationship would have a correlation coefficient of either 1 or -1. But wait, the slope of the line is 1/2, which is positive. So does that mean the correlation coefficient is 1? Because the slope is positive, it's a positive correlation. If the slope were negative, it would be a negative correlation, so the coefficient would be -1. Let me think about this a bit more. The formula for the correlation coefficient, r, is something like the covariance of x and y divided by the product of their standard deviations. But since all the points lie exactly on a straight line, the covariance should be directly related to the slope of that line.Actually, I recall that if y is a linear function of x, then the correlation coefficient r is equal to the sign of the slope. So if the slope is positive, r is 1, and if the slope is negative, r is -1. The magnitude is always 1 because it's a perfect linear relationship. So in this case, since the slope is 1/2, which is positive, the correlation coefficient should be 1. That makes sense because all the points are perfectly aligned with a positive trend. Let me double-check. If I had a set of points that perfectly fit a line with a positive slope, the correlation should be 1. If it were a negative slope, it would be -1. If the points didn't lie on a straight line, the correlation would be somewhere between -1 and 1, depending on how scattered they are. Yeah, I think that's right. So the answer should be D: 1.

💡First, I need to determine the total area of the 4 by 4 grid. Each small square measures 3 cm by 3 cm, so the area of one small square is 9 square cm. Since there are 16 small squares, the total area of the grid is 16 multiplied by 9, which equals 144 square cm.Next, I need to account for the four unshaded circles placed on top of the grid. The problem doesn't specify the size of the circles, so I'll assume each circle has a diameter equal to the side length of two small squares. This means the diameter of each circle is 6 cm, and therefore, the radius is 3 cm.The area of one circle can be calculated using the formula ( pi r^2 ). Substituting the radius of 3 cm, the area of one circle is ( 9pi ) square cm. Since there are four circles, the total unshaded area covered by the circles is ( 4 times 9pi = 36pi ) square cm.Finally, to find the area of the visible shaded region, I subtract the total unshaded area from the total area of the grid:[144 - 36pi]This expression is in the form ( A - Bpi ), where ( A = 144 ) and ( B = 36 ). Therefore, the sum ( A + B ) is:[144 + 36 = 180]

💡To determine the score each junior received, let's break down the problem step by step.1. **Total number of students**: Let ( n ) represent the total number of students in the class. There are ( 0.2n ) juniors and ( 0.8n ) seniors.2. **Total score of all students**: With an overall average score of 85, the total score for all students is ( 85n ).3. **Total score of the seniors**: The seniors' average score is 84, so their total score is ( 84 times 0.8n = 67.2n ).4. **Total score of the juniors**: The total score from all students is ( 85n ), and the seniors' total score is ( 67.2n ). Therefore, the total score for the juniors is ( 85n - 67.2n = 17.8n ).5. **Average score of each junior**: The number of juniors is ( 0.2n ), and their total score is ( 17.8n ). Thus, the average score for each junior is [ frac{17.8n}{0.2n} = 89. ]6. **Conclusion**: Each junior scored 89 on the test.

💡Okay, so I need to find the digit in the hundreds place of (8^{1234}) when it's written out in decimal. Hmm, that seems pretty big. I mean, (8^{1234}) is an enormous number, way too big to compute directly. So, I need a smarter way to figure out just the hundreds digit without calculating the whole thing.First, I remember that when dealing with large exponents and looking for specific digits, modular arithmetic can be really helpful. Specifically, if I want the last three digits of a number, I can compute the number modulo 1000. That will give me the last three digits, and from there, I can easily pick out the hundreds digit.So, my goal is to compute (8^{1234} mod 1000). Once I have that result, the hundreds digit will be the first digit of that three-digit number.But computing (8^{1234}) directly is impossible because it's such a huge number. Instead, I can use Euler's theorem or look for patterns in the powers of 8 modulo 1000. Euler's theorem says that if two numbers are coprime, then (a^{phi(n)} equiv 1 mod n), where (phi(n)) is Euler's totient function. However, 8 and 1000 are not coprime because they share factors of 2. So, Euler's theorem doesn't apply here.Maybe I can break down the modulus into smaller, coprime components using the Chinese Remainder Theorem. 1000 factors into 8 and 125, which are coprime. So, if I can find (8^{1234} mod 8) and (8^{1234} mod 125), I can combine these results to get (8^{1234} mod 1000).Starting with (8^{1234} mod 8): Well, any multiple of 8 modulo 8 is 0. So, (8^{1234} mod 8 = 0).Now, for (8^{1234} mod 125). Since 8 and 125 are coprime (their greatest common divisor is 1), Euler's theorem can be applied here. First, I need to compute (phi(125)). Since 125 is (5^3), (phi(125) = 125 times (1 - 1/5) = 100). So, Euler's theorem tells us that (8^{100} equiv 1 mod 125).Therefore, (8^{1234}) can be written as (8^{100 times 12 + 34}), which simplifies to ((8^{100})^{12} times 8^{34}). Since (8^{100} equiv 1 mod 125), this becomes (1^{12} times 8^{34} mod 125), which is just (8^{34} mod 125).Now, I need to compute (8^{34} mod 125). This is still a bit tedious, but I can use exponentiation by squaring to make it manageable.Let's compute powers of 8 modulo 125 step by step:- (8^1 = 8 mod 125 = 8)- (8^2 = 64 mod 125 = 64)- (8^4 = (8^2)^2 = 64^2 = 4096 mod 125). Let's compute 4096 divided by 125: 125*32=4000, so 4096-4000=96. So, (8^4 equiv 96 mod 125).- (8^8 = (8^4)^2 = 96^2 = 9216 mod 125). 125*73=9125, so 9216-9125=91. Thus, (8^8 equiv 91 mod 125).- (8^{16} = (8^8)^2 = 91^2 = 8281 mod 125). 125*66=8250, so 8281-8250=31. Therefore, (8^{16} equiv 31 mod 125).- (8^{32} = (8^{16})^2 = 31^2 = 961 mod 125). 125*7=875, so 961-875=86. Hence, (8^{32} equiv 86 mod 125).Now, to get (8^{34}), we can write it as (8^{32} times 8^2). We already have (8^{32} equiv 86) and (8^2 = 64).So, (8^{34} equiv 86 times 64 mod 125). Let's compute 86*64:86*60=5160, and 86*4=344, so total is 5160+344=5504.Now, 5504 divided by 125: 125*44=5500, so 5504-5500=4. Therefore, (8^{34} equiv 4 mod 125).So, summarizing:- (8^{1234} equiv 0 mod 8)- (8^{1234} equiv 4 mod 125)Now, we need to find a number x such that:x ≡ 0 mod 8x ≡ 4 mod 125We can express x as 125k + 4 for some integer k. We need this x to also be divisible by 8, so:125k + 4 ≡ 0 mod 8125 mod 8 is 5 (since 8*15=120, 125-120=5), so:5k + 4 ≡ 0 mod 85k ≡ -4 mod 8But -4 mod 8 is 4, so:5k ≡ 4 mod 8Now, we need to solve for k. Multiply both sides by the modular inverse of 5 mod 8. The inverse of 5 mod 8 is 5, because 5*5=25≡1 mod 8.So, k ≡ 4*5 mod 8k ≡ 20 mod 820 mod 8 is 4, so k ≡ 4 mod 8Therefore, k = 8m + 4 for some integer m.Plugging back into x:x = 125k + 4 = 125*(8m +4) +4 = 1000m + 500 +4 = 1000m + 504So, the smallest positive solution is x=504. Therefore, (8^{1234} equiv 504 mod 1000).Thus, the last three digits of (8^{1234}) are 504. Therefore, the hundreds digit is 5.Wait, but earlier I thought the hundreds digit was 1. Hmm, maybe I made a mistake somewhere.Let me double-check my calculations.Starting from (8^{34} mod 125):I had (8^{32} equiv 86), then (8^{34} = 8^{32} * 8^2 = 86*64). 86*64=5504, and 5504 mod 125 is 5504 - 125*44=5504-5500=4. So that part is correct.Then, solving x ≡ 0 mod 8 and x ≡4 mod 125:x=125k +4. 125k +4 ≡5k +4 ≡0 mod8. So, 5k ≡-4 mod8, which is 5k≡4 mod8. Multiplying both sides by 5 inverse, which is 5: k≡20 mod8≡4 mod8. So, k=8m+4.Thus, x=125*(8m+4)+4=1000m +504. So, x≡504 mod1000.Therefore, last three digits are 504, so hundreds digit is 5.Wait, but in the initial assistant answer, it was 1. So, which one is correct?Let me check another way.Alternatively, perhaps the cycle length of 8 modulo 1000 is shorter than 100. Let me see.Compute powers of 8 modulo 1000:8^1=88^2=648^3=5128^4=4096 mod1000=968^5=96*8=7688^6=768*8=6144 mod1000=1448^7=144*8=1152 mod1000=1528^8=152*8=1216 mod1000=2168^9=216*8=1728 mod1000=7288^10=728*8=5824 mod1000=8248^11=824*8=6592 mod1000=5928^12=592*8=4736 mod1000=7368^13=736*8=5888 mod1000=8888^14=888*8=7104 mod1000=1048^15=104*8=8328^16=832*8=6656 mod1000=6568^17=656*8=5248 mod1000=2488^18=248*8=1984 mod1000=9848^19=984*8=7872 mod1000=8728^20=872*8=6976 mod1000=9768^21=976*8=7808 mod1000=8088^22=808*8=6464 mod1000=4648^23=464*8=3712 mod1000=7128^24=712*8=5696 mod1000=6968^25=696*8=5568 mod1000=5688^26=568*8=4544 mod1000=5448^27=544*8=4352 mod1000=3528^28=352*8=2816 mod1000=8168^29=816*8=6528 mod1000=5288^30=528*8=4224 mod1000=2248^31=224*8=1792 mod1000=7928^32=792*8=6336 mod1000=3368^33=336*8=2688 mod1000=6888^34=688*8=5504 mod1000=5048^35=504*8=4032 mod1000=328^36=32*8=2568^37=256*8=2048 mod1000=488^38=48*8=3848^39=384*8=3072 mod1000=728^40=72*8=5768^41=576*8=4608 mod1000=6088^42=608*8=4864 mod1000=8648^43=864*8=6912 mod1000=9128^44=912*8=7296 mod1000=2968^45=296*8=2368 mod1000=3688^46=368*8=2944 mod1000=9448^47=944*8=7552 mod1000=5528^48=552*8=4416 mod1000=4168^49=416*8=3328 mod1000=3288^50=328*8=2624 mod1000=6248^51=624*8=4992 mod1000=9928^52=992*8=7936 mod1000=9368^53=936*8=7488 mod1000=4888^54=488*8=3904 mod1000=9048^55=904*8=7232 mod1000=2328^56=232*8=1856 mod1000=8568^57=856*8=6848 mod1000=8488^58=848*8=6784 mod1000=7848^59=784*8=6272 mod1000=2728^60=272*8=2176 mod1000=1768^61=176*8=1408 mod1000=4088^62=408*8=3264 mod1000=2648^63=264*8=2112 mod1000=1128^64=112*8=8968^65=896*8=7168 mod1000=1688^66=168*8=1344 mod1000=3448^67=344*8=2752 mod1000=7528^68=752*8=6016 mod1000=168^69=16*8=1288^70=128*8=1024 mod1000=248^71=24*8=1928^72=192*8=1536 mod1000=5368^73=536*8=4288 mod1000=2888^74=288*8=2304 mod1000=3048^75=304*8=2432 mod1000=4328^76=432*8=3456 mod1000=4568^77=456*8=3648 mod1000=6488^78=648*8=5184 mod1000=1848^79=184*8=1472 mod1000=4728^80=472*8=3776 mod1000=7768^81=776*8=6208 mod1000=2088^82=208*8=1664 mod1000=6648^83=664*8=5312 mod1000=3128^84=312*8=2496 mod1000=4968^85=496*8=3968 mod1000=9688^86=968*8=7744 mod1000=7448^87=744*8=5952 mod1000=9528^88=952*8=7616 mod1000=6168^89=616*8=4928 mod1000=9288^90=928*8=7424 mod1000=4248^91=424*8=3392 mod1000=3928^92=392*8=3136 mod1000=1368^93=136*8=1088 mod1000=888^94=88*8=7048^95=704*8=5632 mod1000=6328^96=632*8=5056 mod1000=568^97=56*8=4488^98=448*8=3584 mod1000=5848^99=584*8=4672 mod1000=6728^100=672*8=5376 mod1000=376Wait, so at 8^100, we get 376 mod1000, not 1. That contradicts Euler's theorem, which said that 8^100 ≡1 mod125. But here, 8^100 mod1000 is 376, which is not 1. So, perhaps my earlier assumption was wrong.Wait, actually, Euler's theorem applies to modulus where the base and modulus are coprime. 8 and 125 are coprime, so 8^100 ≡1 mod125, but 8 and 1000 are not coprime, so Euler's theorem doesn't apply to modulus 1000.So, in my earlier approach, I correctly split the modulus into 8 and 125, computed modulo 125, and then used the Chinese Remainder Theorem. So, that part should be correct.But when I computed 8^34 mod125, I got 4, and then solving the congruence, I got x=504 mod1000, so last three digits 504, hundreds digit 5.But when I computed the cycle manually, I saw that 8^34 mod1000 is 504, which matches. So, that seems consistent.But in the initial assistant answer, it was 1. So, perhaps the initial answer was incorrect.Wait, let me check the initial assistant's answer again.They listed the cycle of last three digits as repeating every 20 numbers, with 8^20 ending with 976, and then 8^21=808, etc., and then for 8^1234, they said 1234 mod20=14, so last three digits 104, hundreds digit 1.But according to my calculations, 8^34 mod1000=504, and 1234 mod100=34, so 8^1234 mod1000=504, hundreds digit 5.So, which is correct?Wait, perhaps the cycle length is 100, not 20. Because when I computed up to 8^100, it was 376 mod1000, not back to 008. So, the cycle length is longer.In fact, the multiplicative order of 8 modulo 1000 is the smallest k such that 8^k ≡1 mod1000. But since 8 and 1000 are not coprime, 8^k can never be 1 mod1000. Therefore, the cycle doesn't repeat in the same way.Instead, the cycle of the last three digits of 8^n is periodic, but not with a period that brings it back to 008. Instead, it's a different kind of cycle.So, perhaps the initial assistant answer was incorrect in assuming a cycle of 20.Alternatively, perhaps I made a mistake in my calculations.Wait, let's see. The initial assistant answer said that 8^20 mod1000=976, and then 8^21=808, etc., and then for 8^1234, since 1234 mod20=14, so last three digits 104, hundreds digit 1.But according to my manual computation, 8^14 mod1000=104, which is correct. So, if the cycle is 20, then 8^(20k + r) mod1000=8^r mod1000.But in my earlier approach, using Chinese Remainder Theorem, I got 504, which is different.So, which one is correct?Wait, let's compute 8^14 mod1000=104, as per the initial assistant.But according to my manual computation, 8^34 mod1000=504.Wait, 1234 divided by 20 is 61 with a remainder of 14, so 8^1234 mod1000=8^14 mod1000=104.But according to my Chinese Remainder Theorem approach, I got 504.So, there's a discrepancy here.Wait, perhaps the cycle length is indeed 100, not 20. Because when I computed up to 8^100, it was 376, not back to 008.So, perhaps the initial assistant answer was incorrect in assuming a cycle of 20.Alternatively, perhaps the cycle length is 100, but with a different behavior.Wait, let's check 8^20 mod1000=976, as per the initial assistant.8^21=808, 8^22=464, 8^23=712, 8^24=696, 8^25=568, 8^26=544, 8^27=352, 8^28=816, 8^29=528, 8^30=224, 8^31=792, 8^32=336, 8^33=688, 8^34=504, 8^35=32, 8^36=256, 8^37=48, 8^38=384, 8^39=72, 8^40=576, 8^41=608, 8^42=864, 8^43=912, 8^44=296, 8^45=368, 8^46=944, 8^47=552, 8^48=416, 8^49=328, 8^50=624, 8^51=992, 8^52=936, 8^53=488, 8^54=904, 8^55=232, 8^56=856, 8^57=848, 8^58=784, 8^59=272, 8^60=176, 8^61=408, 8^62=264, 8^63=112, 8^64=896, 8^65=168, 8^66=344, 8^67=752, 8^68=16, 8^69=128, 8^70=24, 8^71=192, 8^72=536, 8^73=288, 8^74=304, 8^75=432, 8^76=456, 8^77=648, 8^78=184, 8^79=472, 8^80=776, 8^81=208, 8^82=664, 8^83=312, 8^84=496, 8^85=968, 8^86=744, 8^87=952, 8^88=616, 8^89=928, 8^90=424, 8^91=392, 8^92=136, 8^93=88, 8^94=704, 8^95=632, 8^96=56, 8^97=448, 8^98=584, 8^99=672, 8^100=376.So, at 8^100, it's 376, not 008. So, the cycle length is not 100 either.Wait, but in my Chinese Remainder Theorem approach, I got 504, which is 8^34 mod1000.But 1234 mod100=34, so 8^1234 mod1000=8^34 mod1000=504.But according to the initial assistant, 1234 mod20=14, so 8^14 mod1000=104.So, which one is correct?Wait, perhaps the cycle length is 100, but the initial assistant assumed a cycle of 20, which is incorrect.Alternatively, perhaps the cycle length is indeed 100, but the initial assistant made a mistake.Wait, let's compute 8^100 mod1000=376, as above.Then, 8^200=(8^100)^2=376^2=141376 mod1000=376.So, 8^200 mod1000=376.Similarly, 8^300=376*8^100=376*376=141376 mod1000=376.So, it seems that 8^100 mod1000=376, and 8^(100k) mod1000=376 for any k>=1.Therefore, the cycle length is not 100, because 8^100 is not 1 mod1000.Instead, the multiplicative order is undefined because 8 and 1000 are not coprime.Therefore, the cycle of the last three digits of 8^n is not periodic in the same way as when the base and modulus are coprime.Therefore, the initial assistant's assumption of a cycle length of 20 is incorrect.So, my approach using Chinese Remainder Theorem is more accurate.Therefore, the correct last three digits are 504, so the hundreds digit is 5.But wait, in my manual computation, 8^34 mod1000=504, which matches the Chinese Remainder Theorem result.Therefore, the correct hundreds digit is 5, not 1.So, the initial assistant answer was incorrect in assuming a cycle length of 20.Therefore, the correct answer is 5.

💡Okay, so I'm trying to solve this problem about a geometric sequence. Let me read it again and make sure I understand what's being asked.We have a geometric sequence {a_n}. The first piece of information is that a₂ multiplied by a₃ equals 2a₁. So, a₂ * a₃ = 2a₁. Then, the arithmetic mean of a₄ and 2a₇ is 17. So, that means (a₄ + 2a₇)/2 = 17, which simplifies to a₄ + 2a₇ = 34.Next, we're told to define a new sequence {b_n} where each term is b_n = a_{2n-1} - a_{2n}. We need to find the sum of the first 2n terms of this sequence {b_n}.Alright, so let's break this down step by step.First, since {a_n} is a geometric sequence, I know that each term can be expressed as a_n = a₁ * q^{n-1}, where a₁ is the first term and q is the common ratio.Given that, let's write out the terms a₂, a₃, a₄, and a₇ in terms of a₁ and q.- a₂ = a₁ * q^{2-1} = a₁ * q- a₃ = a₁ * q^{3-1} = a₁ * q²- a₄ = a₁ * q^{4-1} = a₁ * q³- a₇ = a₁ * q^{7-1} = a₁ * q⁶Now, let's plug these into the given equations.First equation: a₂ * a₃ = 2a₁Substituting the expressions we have:(a₁ * q) * (a₁ * q²) = 2a₁Simplify the left side:a₁² * q³ = 2a₁We can divide both sides by a₁ (assuming a₁ ≠ 0, which makes sense because otherwise the sequence would be trivial):a₁ * q³ = 2So, that's our first equation: a₁ * q³ = 2. Let's keep that in mind.Second piece of information: The arithmetic mean of a₄ and 2a₇ is 17. As I mentioned earlier, that translates to (a₄ + 2a₇)/2 = 17, so multiplying both sides by 2 gives a₄ + 2a₇ = 34.Substituting the expressions for a₄ and a₇:a₁ * q³ + 2*(a₁ * q⁶) = 34Simplify:a₁ * q³ + 2a₁ * q⁶ = 34Hmm, from the first equation, we have a₁ * q³ = 2. Maybe we can substitute that into this equation.Let me denote a₁ * q³ as 2, so substituting into the second equation:2 + 2a₁ * q⁶ = 34Subtract 2 from both sides:2a₁ * q⁶ = 32Divide both sides by 2:a₁ * q⁶ = 16Now, we have two equations:1. a₁ * q³ = 22. a₁ * q⁶ = 16Hmm, I can see that the second equation is just the first equation multiplied by q³. Let me check:From equation 1: a₁ * q³ = 2Multiply both sides by q³: a₁ * q⁶ = 2q³But from equation 2, a₁ * q⁶ = 16So, 2q³ = 16Divide both sides by 2:q³ = 8So, q³ = 8. Taking the cube root of both sides, we get q = 2.Now that we have q, we can find a₁ from equation 1:a₁ * q³ = 2We know q³ = 8, so:a₁ * 8 = 2Divide both sides by 8:a₁ = 2/8 = 1/4So, the first term a₁ is 1/4, and the common ratio q is 2.Therefore, the general term of the geometric sequence is:a_n = a₁ * q^{n-1} = (1/4) * 2^{n-1}Simplify that:2^{n-1} is equal to 2^{n} / 2, so:a_n = (1/4) * (2^{n} / 2) = (1/4) * (2^{n-1}) = 2^{n-3}Wait, let me check that again.Wait, 2^{n-1} is 2^{n} / 2, so:a_n = (1/4) * (2^{n} / 2) = (1/4) * (2^{n-1}) = 2^{n-3}Yes, that's correct because 2^{n-1} divided by 4 is 2^{n-1} / 2² = 2^{n-3}.So, a_n = 2^{n-3}.Alright, now that we have the general term, let's move on to the sequence {b_n}.We are told that b_n = a_{2n-1} - a_{2n}.So, let's write expressions for a_{2n-1} and a_{2n}.First, a_{2n-1} = 2^{(2n - 1) - 3} = 2^{2n - 4}Similarly, a_{2n} = 2^{2n - 3}Therefore, b_n = a_{2n-1} - a_{2n} = 2^{2n - 4} - 2^{2n - 3}Let me factor out 2^{2n - 4}:b_n = 2^{2n - 4} (1 - 2) = 2^{2n - 4} (-1) = -2^{2n - 4}So, each term b_n is equal to -2^{2n - 4}.Now, we need to find the sum of the first 2n terms of the sequence {b_n}.Wait, hold on. The sequence {b_n} is defined for n ∈ ℕ*, so n = 1, 2, 3, etc. So, the first 2n terms would be b₁, b₂, ..., b_{2n}.But let me think about whether that's correct. Wait, no, actually, the problem says "the sum of the first 2n terms of the sequence {b_n}". So, if we're considering the sequence {b_n}, then the first term is b₁, the second is b₂, up to b_{2n}.But wait, let me check the problem statement again: "Find the sum of the first 2n terms of the sequence {b_n}." So, yes, it's the sum from k=1 to k=2n of b_k.But let's see, since b_n is defined as a_{2n-1} - a_{2n}, each term b_n is the difference between two consecutive terms of the original geometric sequence, but specifically the odd and even terms.Wait, but perhaps it's better to consider the sum of the first 2n terms of {b_n}, which would be the sum from k=1 to 2n of b_k.But let me compute b_n first.We have b_n = -2^{2n - 4}.So, let's write out the first few terms to see the pattern.For n=1: b₁ = -2^{2*1 - 4} = -2^{-2} = -1/4n=2: b₂ = -2^{2*2 - 4} = -2^{0} = -1n=3: b₃ = -2^{2*3 - 4} = -2^{2} = -4n=4: b₄ = -2^{4} = -16Wait, that seems like each term is multiplied by 4 each time. Let me check:From b₁ to b₂: -1/4 to -1, which is multiplied by 4.From b₂ to b₃: -1 to -4, multiplied by 4.From b₃ to b₄: -4 to -16, multiplied by 4.Yes, so the sequence {b_n} is a geometric sequence with first term b₁ = -1/4 and common ratio r = 4.Wait, but let me confirm that.Wait, the general term is b_n = -2^{2n - 4}.Let me write that as b_n = -2^{2(n - 2)} = - (2^2)^{n - 2} = -4^{n - 2}.Alternatively, we can write it as b_n = (-1/4) * 4^{n - 1}.Because 4^{n - 1} is 4^{n - 2} * 4, so:-4^{n - 2} = (-1/4) * 4^{n - 1}.Yes, that's correct. So, the first term when n=1 is (-1/4)*4^{0} = -1/4, which matches b₁.Similarly, for n=2: (-1/4)*4^{1} = -1, which is b₂.So, the sequence {b_n} is indeed a geometric sequence with first term a = -1/4 and common ratio r = 4.Now, we need to find the sum of the first 2n terms of this sequence.The formula for the sum of the first m terms of a geometric sequence is S_m = a*(1 - r^m)/(1 - r), provided that r ≠ 1.In this case, a = -1/4, r = 4, and m = 2n.So, plugging into the formula:S_{2n} = (-1/4)*(1 - 4^{2n}) / (1 - 4)Simplify the denominator: 1 - 4 = -3.So, S_{2n} = (-1/4)*(1 - 4^{2n}) / (-3)The negatives in numerator and denominator will cancel out:= (-1/4)*(1 - 4^{2n}) / (-3) = (1/4)*(1 - 4^{2n}) / 3Simplify:= (1 - 4^{2n}) / 12So, the sum of the first 2n terms of {b_n} is (1 - 4^{2n}) / 12.Wait, but let me check the calculation again.Wait, S_{2n} = a*(1 - r^{2n}) / (1 - r)a = -1/4, r = 4.So,S_{2n} = (-1/4)*(1 - 4^{2n}) / (1 - 4) = (-1/4)*(1 - 4^{2n}) / (-3)Multiplying numerator and denominator:= [(-1/4) * (1 - 4^{2n})] / (-3) = [(-1/4) / (-3)] * (1 - 4^{2n})Which is (1/12)*(1 - 4^{2n})Yes, that's correct.Alternatively, since 4^{2n} is (4^2)^n = 16^n, but I think 4^{2n} is fine as it is.So, the sum is (1 - 4^{2n}) / 12.Alternatively, we can write it as (1 - 16^n)/12, but 4^{2n} is more straightforward.Wait, but let me check if the initial steps were correct.We found that a_n = 2^{n - 3}, correct?Then, b_n = a_{2n - 1} - a_{2n} = 2^{(2n - 1) - 3} - 2^{2n - 3} = 2^{2n - 4} - 2^{2n - 3} = 2^{2n - 4}(1 - 2) = -2^{2n - 4}Yes, that's correct.So, b_n = -2^{2n - 4} = - (2^{2n} / 2^4) = - (4^n / 16) = - (4^{n} / 16) = - (1/16) * 4^nWait, that's another way to write it. So, b_n = - (1/16) * 4^nBut earlier, we saw that b_n = (-1/4) * 4^{n - 1}Let me check if these are equivalent.(-1/4) * 4^{n - 1} = (-1/4) * (4^{n} / 4) ) = (-1/4) * (4^{n}/4) = (-1/16) * 4^{n}Yes, so both expressions are equivalent.So, whether we write it as (-1/4) * 4^{n - 1} or - (1/16) * 4^n, it's the same.But for the sum, it's perhaps better to express it in terms of 4^{n}.But in any case, the sum formula gives us (1 - 4^{2n}) / 12.Wait, let me check with n=1.If n=1, then the sum of the first 2 terms of {b_n} is b₁ + b₂.From earlier, b₁ = -1/4, b₂ = -1.So, sum is -1/4 + (-1) = -5/4.Now, let's plug n=1 into our formula:(1 - 4^{2*1}) / 12 = (1 - 16)/12 = (-15)/12 = -5/4, which matches.Similarly, for n=2, sum of first 4 terms: b₁ + b₂ + b₃ + b₄.From earlier, b₁ = -1/4, b₂ = -1, b₃ = -4, b₄ = -16.Sum: -1/4 -1 -4 -16 = (-1/4 -1) = -5/4; (-5/4 -4) = -21/4; (-21/4 -16) = (-21/4 -64/4) = -85/4.Now, plug n=2 into our formula:(1 - 4^{4}) / 12 = (1 - 256)/12 = (-255)/12 = -85/4, which matches.So, the formula seems correct.Therefore, the sum of the first 2n terms of {b_n} is (1 - 4^{2n}) / 12.Alternatively, since 4^{2n} is (4^2)^n = 16^n, we can write it as (1 - 16^n)/12, but I think 4^{2n} is acceptable.So, the final answer is (1 - 4^{2n}) / 12.Wait, but in the initial problem statement, the user provided a solution that ended up with the same answer, but let me make sure I didn't make any mistakes.Wait, in the initial problem, the user wrote:"Thus, a_n = a₁ q^{n-1} = 2^{n-3}, Therefore, b_n = a_{2n-1} - a_{2n} = 2^{2n-3}/2 - 2^{2n-3} = -2^{2n-4}, Then, the sum of the first 2n terms is T_{2n} = (-1/4)(1 - 4^{2n}) / (1 - 4) = (1/12)(1 - 4^{2n})"Wait, that's exactly what I did, so I think my answer is correct.Therefore, the sum is (1 - 4^{2n}) / 12, which can be written as boxed{dfrac{1 - 4^{2n}}{12}}.Alternatively, if we factor out a negative, it's the same as boxed{dfrac{1}{12}(1 - 4^{2n})}.Either way is correct, but I think the first form is more standard.**Final Answer**The sum of the first (2n) terms of the sequence ({b_n}) is boxed{dfrac{1 - 4^{2n}}{12}}.