💡First, I need to evaluate each option one by one to determine which calculations are correct.**Option A:** The expression is ( a^{8} div a^{4} ). According to the division rule of powers with the same base, this should be ( a^{8-4} = a^{4} ). However, the given result is ( a^{2} ), which is incorrect.**Option B:** The expression is ( (a^{2})^{3} ). Using the rule for raising a power to a power, this should be ( a^{2 times 3} = a^{6} ). The given result is ( a^{5} ), which is incorrect.**Option C:** The expression is ( (3a)^{3} ). Applying the rules for raising a product to a power, this should be ( 3^{3} times a^{3} = 27a^{3} ). The given result is ( 9a^{3} ), which is incorrect.**Option D:** The expression is ( (-a)^{3} cdot (-a)^{5} ). Using the multiplication rule of powers with the same base, this becomes ( (-a)^{3+5} = (-a)^{8} ). Since any even power of a negative number is positive, this simplifies to ( a^{8} ), which matches the given result. Therefore, this option is correct.After evaluating all options, only Option D is correct.

💡Okay, so I have this inequality to solve: (2 - x)/3 + 2 > x - (x - 2)/2. Hmm, it's a bit intimidating at first glance, but I think I can break it down step by step. Let me try to remember how to handle inequalities with fractions. I think the key is to eliminate the denominators to make it simpler.First, I'll look at both sides of the inequality. On the left side, there's (2 - x)/3 plus 2, and on the right side, there's x minus (x - 2)/2. Maybe I should simplify each side separately before trying to solve the inequality.Starting with the left side: (2 - x)/3 + 2. I can rewrite 2 as 6/3 to have a common denominator. So, that becomes (2 - x)/3 + 6/3, which is (2 - x + 6)/3. Combining like terms in the numerator gives me (8 - x)/3. Okay, that's simpler.Now, the right side: x - (x - 2)/2. I think I should also combine these terms. Maybe I can rewrite x as 2x/2 to have a common denominator. So, that would be 2x/2 - (x - 2)/2. Now, combining the numerators: 2x - (x - 2) all over 2. Distribute the negative sign: 2x - x + 2 over 2, which simplifies to (x + 2)/2.So now, the inequality is (8 - x)/3 > (x + 2)/2. To eliminate the denominators, I can find the least common denominator (LCD) of 3 and 2, which is 6. I'll multiply both sides of the inequality by 6 to get rid of the fractions. Multiplying the left side: 6 * (8 - x)/3 = 2*(8 - x) = 16 - 2x. Multiplying the right side: 6 * (x + 2)/2 = 3*(x + 2) = 3x + 6. So now, the inequality becomes 16 - 2x > 3x + 6.Next, I need to get all the x terms on one side and the constants on the other. I'll start by adding 2x to both sides to get rid of the -2x on the left. That gives me 16 > 5x + 6. Now, I'll subtract 6 from both sides to isolate the term with x: 16 - 6 > 5x, which simplifies to 10 > 5x.To solve for x, I'll divide both sides by 5: 10/5 > x, which simplifies to 2 > x. In other words, x < 2.Wait, did I remember to flip the inequality sign when dividing by a negative? Let me check. I divided by 5, which is positive, so the inequality sign stays the same. If I had multiplied or divided by a negative number, I would have flipped the sign, but since 5 is positive, I don't need to change it. So, x is less than 2.Just to make sure I didn't make any mistakes, I'll test a value less than 2, say x = 1, in the original inequality. Plugging in x = 1:Left side: (2 - 1)/3 + 2 = (1)/3 + 2 ≈ 0.333 + 2 = 2.333Right side: 1 - (1 - 2)/2 = 1 - (-1)/2 = 1 + 0.5 = 1.5So, 2.333 > 1.5, which is true. Now, let's test a value greater than 2, say x = 3:Left side: (2 - 3)/3 + 2 = (-1)/3 + 2 ≈ -0.333 + 2 = 1.666Right side: 3 - (3 - 2)/2 = 3 - (1)/2 = 3 - 0.5 = 2.5So, 1.666 > 2.5 is false. That confirms that x must be less than 2 for the inequality to hold.I think I've got it right. The solution is x < 2.

💡Okay, so I have this problem here: find the integer x that satisfies the equation 10x + 3 ≡ 7 mod 18. Hmm, modular arithmetic, right? I remember that congruences are kind of like equations, but everything is modulo some number, which in this case is 18. So, I need to solve for x in this congruence.Let me write down the equation again to make sure I have it right: 10x + 3 ≡ 7 (mod 18). I think the first step is to isolate the term with x. That usually makes things easier. So, if I subtract 3 from both sides, I can get rid of that constant term on the left. Let me do that:10x + 3 - 3 ≡ 7 - 3 (mod 18)Simplifying both sides, that becomes:10x ≡ 4 (mod 18)Alright, so now I have 10x ≡ 4 mod 18. I need to solve for x. I remember that to solve equations like ax ≡ b mod m, you can sometimes divide both sides by a, but only if a and m are coprime, meaning their greatest common divisor (GCD) is 1. If they aren't coprime, you have to adjust the modulus accordingly.So, let me check the GCD of 10 and 18. The factors of 10 are 1, 2, 5, 10, and the factors of 18 are 1, 2, 3, 6, 9, 18. The common factors are 1 and 2, so the GCD is 2. Since the GCD is 2, which isn't 1, I can't just divide both sides by 10 directly. Instead, I think I need to divide the entire congruence by 2, which will also divide the modulus.Let me try that. If I divide both sides of 10x ≡ 4 mod 18 by 2, I get:(10/2)x ≡ (4/2) mod (18/2)Which simplifies to:5x ≡ 2 mod 9Okay, so now I have 5x ≡ 2 mod 9. This is better because 5 and 9 are coprime (their GCD is 1). That means I can find the multiplicative inverse of 5 modulo 9, and then multiply both sides by that inverse to solve for x.I need to find an integer y such that 5y ≡ 1 mod 9. Let me think about what y could be. Let's try plugging in some numbers:5*1 = 5 ≡ 5 mod 9 → Not 15*2 = 10 ≡ 1 mod 9 → Oh, that works!So, the multiplicative inverse of 5 mod 9 is 2. Great, now I can multiply both sides of the equation 5x ≡ 2 mod 9 by 2:2*5x ≡ 2*2 mod 9Simplifying, that becomes:10x ≡ 4 mod 9Wait, but 10x mod 9 is the same as (10 mod 9)x, which is 1x, so:x ≡ 4 mod 9So, x is congruent to 4 modulo 9. That means the solutions for x are all integers that are 4 more than a multiple of 9. In other words, x can be written as 9k + 4 for some integer k.But the problem just asks for the integer x that satisfies the equation. Since modular equations have infinitely many solutions, but they all differ by multiples of the modulus, I think the answer is expected to be the smallest positive integer solution, which would be 4.Wait, let me double-check to make sure I didn't make a mistake. Let's plug x = 4 back into the original equation:10*4 + 3 = 40 + 3 = 43Now, 43 mod 18 is 43 - 2*18 = 43 - 36 = 7. So, 43 ≡ 7 mod 18, which matches the original equation. Perfect, that works.Just to be thorough, let me check another solution, say x = 4 + 9 = 13.10*13 + 3 = 130 + 3 = 133133 mod 18: 18*7 = 126, so 133 - 126 = 7. So, 133 ≡ 7 mod 18. That also works. So, x = 13 is another solution, but since the problem didn't specify a range, the smallest positive integer solution is 4.I think that's it. I followed the steps: simplified the equation, divided by the GCD, found the inverse, solved for x, and verified the solution. It all checks out.

💡First, identify the height of the old lamp, which is 1 foot.Next, determine the additional height of the new lamp compared to the old lamp, which is 1.33 feet.Add the additional height to the height of the old lamp to find the total height of the new lamp.1 foot + 1.33 feet = 2.33 feetTherefore, the new lamp is 2.33 feet tall.

💡First, I'll define the amounts that A, B, and C have as A, B, and C respectively.From the problem, I have three pieces of information:1. The total amount that A, B, and C have together is Rs. 700.2. A and C together have Rs. 300.3. B and C together have Rs. 600.I can set up these statements as equations:1. A + B + C = 7002. A + C = 3003. B + C = 600To find out how much C has, I'll solve these equations step by step.First, from equation (2), I can express A in terms of C:A = 300 - CNext, I'll substitute this expression for A into equation (1):(300 - C) + B + C = 700Simplifying this, the -C and +C cancel out, leaving:300 + B = 700Solving for B:B = 700 - 300B = 400Now that I know B is 400, I'll substitute this value into equation (3):400 + C = 600Solving for C:C = 600 - 400C = 200Therefore, C has Rs. 200.

💡Okay, so I have this problem about an arithmetic sequence. Let me try to understand what it's asking. It says that there's an arithmetic sequence {a_n} with a non-zero common difference. That means each term increases by a fixed amount, right? So, if I denote the first term as a_1 and the common difference as d, then the nth term can be written as a_n = a_1 + (n-1)d. The problem gives me two pieces of information. First, a_3 = 7. That should help me find a relationship between a_1 and d. Second, it says that a_1, a_4, and a_{13} form a geometric sequence. Hmm, okay, so a geometric sequence means each term is multiplied by a common ratio. So, if I have three terms in a geometric sequence, the square of the middle term should be equal to the product of the first and the third terms. That gives me another equation to work with.Let me write down what I know:1. a_3 = 7. Since it's an arithmetic sequence, a_3 = a_1 + 2d. So, equation one is: a_1 + 2d = 7.2. a_1, a_4, a_{13} form a geometric sequence. So, in a geometric sequence, the ratio between consecutive terms is constant. Therefore, (a_4)^2 = a_1 * a_{13}.Let me express a_4 and a_{13} in terms of a_1 and d.a_4 = a_1 + 3da_{13} = a_1 + 12dSo, plugging these into the geometric sequence condition:(a_1 + 3d)^2 = a_1 * (a_1 + 12d)Okay, so now I have two equations:1. a_1 + 2d = 72. (a_1 + 3d)^2 = a_1(a_1 + 12d)I need to solve these two equations to find a_1 and d.Let me start with equation 1: a_1 = 7 - 2dNow, substitute this into equation 2:( (7 - 2d) + 3d )^2 = (7 - 2d)( (7 - 2d) + 12d )Simplify the expressions inside the parentheses:Left side: (7 - 2d + 3d) = (7 + d)So, left side becomes (7 + d)^2Right side: (7 - 2d)(7 - 2d + 12d) = (7 - 2d)(7 + 10d)So, expanding both sides:Left side: (7 + d)^2 = 49 + 14d + d^2Right side: (7 - 2d)(7 + 10d) = 7*7 + 7*10d - 2d*7 - 2d*10d = 49 + 70d - 14d - 20d^2 = 49 + 56d - 20d^2So, setting left side equal to right side:49 + 14d + d^2 = 49 + 56d - 20d^2Let me bring all terms to one side:49 + 14d + d^2 - 49 - 56d + 20d^2 = 0Simplify:(49 - 49) + (14d - 56d) + (d^2 + 20d^2) = 0Which simplifies to:0 - 42d + 21d^2 = 0So, 21d^2 - 42d = 0Factor out 21d:21d(d - 2) = 0So, 21d = 0 or d - 2 = 0But the problem states that the common difference is non-zero, so d ≠ 0. Therefore, d = 2.Now, substitute d = 2 back into equation 1: a_1 + 2d = 7So, a_1 + 4 = 7 => a_1 = 3Therefore, the general formula for the arithmetic sequence is:a_n = a_1 + (n - 1)d = 3 + (n - 1)*2 = 3 + 2n - 2 = 2n + 1So, a_n = 2n + 1.Alright, that takes care of part (1). Now, moving on to part (2). It says: Let the sum of the first n terms of the sequence {a_n · 2^n} be S_n, find S_n.So, the sequence is a_n multiplied by 2^n, and we need to find the sum of the first n terms of this new sequence.Given that a_n = 2n + 1, so the sequence becomes (2n + 1) * 2^n.So, S_n = sum_{k=1}^n (2k + 1) * 2^kHmm, okay. So, I need to compute S_n = 3*2^1 + 5*2^2 + 7*2^3 + ... + (2n + 1)*2^nThis looks like a sum that can be tackled using the method of differences or perhaps recognizing it as a combination of geometric series.Let me write out the terms:S_n = 3*2 + 5*4 + 7*8 + ... + (2n + 1)*2^nI notice that each term is of the form (2k + 1)*2^k. Let me try to split this into two separate sums:S_n = sum_{k=1}^n (2k * 2^k) + sum_{k=1}^n (1 * 2^k)So, S_n = 2 * sum_{k=1}^n k*2^k + sum_{k=1}^n 2^kI know that sum_{k=1}^n 2^k is a geometric series. The sum of a geometric series from k=1 to n is 2(2^n - 1)/(2 - 1) = 2^{n+1} - 2.Wait, let me verify that:sum_{k=1}^n 2^k = 2 + 4 + 8 + ... + 2^n = 2(2^n - 1)/(2 - 1) = 2^{n+1} - 2. Yes, that's correct.So, the second sum is 2^{n+1} - 2.Now, the first sum is 2 * sum_{k=1}^n k*2^k. I need a formula for sum_{k=1}^n k*r^k. I remember that there is a formula for this.The general formula for sum_{k=1}^n k*r^k is r(1 - (n+1)r^n + n r^{n+1}) / (1 - r)^2, for r ≠ 1.In this case, r = 2, so plugging in:sum_{k=1}^n k*2^k = 2(1 - (n+1)2^n + n*2^{n+1}) / (1 - 2)^2Simplify denominator: (1 - 2)^2 = 1So, numerator: 2(1 - (n+1)2^n + n*2^{n+1}) = 2 - 2(n+1)2^n + 2n*2^{n+1}Wait, let me compute this step by step.Wait, actually, the formula is:sum_{k=1}^n k r^k = r(1 - (n+1) r^n + n r^{n+1}) / (1 - r)^2So, plugging in r = 2:sum_{k=1}^n k*2^k = 2(1 - (n+1)2^n + n*2^{n+1}) / (1 - 2)^2Compute denominator: (1 - 2)^2 = 1So, numerator: 2(1 - (n+1)2^n + n*2^{n+1})Let me expand this:2*1 - 2*(n+1)2^n + 2*n*2^{n+1}Simplify each term:2 - 2(n+1)2^n + 2n*2^{n+1}Note that 2^{n+1} = 2*2^n, so 2n*2^{n+1} = 2n*2*2^n = 4n*2^nSimilarly, 2(n+1)2^n = 2n*2^n + 2*2^nSo, putting it all together:2 - [2n*2^n + 2*2^n] + 4n*2^nCombine like terms:2 - 2n*2^n - 2*2^n + 4n*2^nCombine the terms with 2^n:(-2n*2^n + 4n*2^n) + (-2*2^n) + 2Which is:(2n*2^n) - 2*2^n + 2Factor out 2^n:2^n(2n - 2) + 2So, sum_{k=1}^n k*2^k = 2^n(2n - 2) + 2Therefore, going back to S_n:S_n = 2 * sum_{k=1}^n k*2^k + sum_{k=1}^n 2^k= 2*(2^n(2n - 2) + 2) + (2^{n+1} - 2)Let me compute each part:First term: 2*(2^n(2n - 2) + 2) = 2*2^n(2n - 2) + 4 = 2^{n+1}(2n - 2) + 4Second term: 2^{n+1} - 2So, adding them together:2^{n+1}(2n - 2) + 4 + 2^{n+1} - 2Combine like terms:2^{n+1}(2n - 2 + 1) + (4 - 2)Simplify:2^{n+1}(2n - 1) + 2So, S_n = (2n - 1)2^{n+1} + 2Wait, let me check that again.Wait, 2^{n+1}(2n - 2) + 2^{n+1} = 2^{n+1}(2n - 2 + 1) = 2^{n+1}(2n - 1)And 4 - 2 = 2So, yes, S_n = (2n - 1)2^{n+1} + 2But let me see if this can be simplified further or perhaps written differently.Alternatively, perhaps I made a miscalculation earlier. Let me verify:sum_{k=1}^n k*2^k = 2(1 - (n+1)2^n + n*2^{n+1}) / (1 - 2)^2Wait, denominator is (1 - 2)^2 = 1, so numerator is 2(1 - (n+1)2^n + n*2^{n+1})Which is 2 - 2(n+1)2^n + 2n*2^{n+1}Wait, 2n*2^{n+1} is 2n*2*2^n = 4n*2^nSo, 2 - 2(n+1)2^n + 4n*2^n= 2 + [ -2(n+1) + 4n ]2^n= 2 + [ -2n - 2 + 4n ]2^n= 2 + (2n - 2)2^nSo, sum_{k=1}^n k*2^k = 2 + (2n - 2)2^nTherefore, 2 * sum_{k=1}^n k*2^k = 2*(2 + (2n - 2)2^n) = 4 + (4n - 4)2^nAnd sum_{k=1}^n 2^k = 2^{n+1} - 2So, S_n = 4 + (4n - 4)2^n + 2^{n+1} - 2Simplify:4 - 2 = 2(4n - 4)2^n + 2^{n+1} = (4n - 4 + 2)2^n = (4n - 2)2^nWait, no:Wait, 2^{n+1} = 2*2^n, so:(4n - 4)2^n + 2*2^n = (4n - 4 + 2)2^n = (4n - 2)2^nSo, S_n = 2 + (4n - 2)2^nFactor out 2 from the second term:= 2 + 2(2n - 1)2^n= 2 + (2n - 1)2^{n+1}Alternatively, factor 2^{n+1}:= (2n - 1)2^{n+1} + 2Hmm, that seems consistent with what I had earlier.Alternatively, perhaps I can factor it differently.Wait, let me see:S_n = (2n - 1)2^{n+1} + 2Alternatively, factor 2^{n+1}:= 2^{n+1}(2n - 1) + 2Alternatively, factor 2:= 2( (2n - 1)2^n + 1 )But perhaps the first form is better.Wait, let me check with small n to see if the formula is correct.Let me test n=1:S_1 = (2*1 + 1)*2^1 = 3*2 = 6Using the formula: (2*1 - 1)2^{2} + 2 = (2 - 1)*4 + 2 = 1*4 + 2 = 6. Correct.n=2:S_2 = 3*2 + 5*4 = 6 + 20 = 26Formula: (2*2 - 1)2^{3} + 2 = (4 - 1)*8 + 2 = 3*8 + 2 = 24 + 2 = 26. Correct.n=3:S_3 = 3*2 + 5*4 + 7*8 = 6 + 20 + 56 = 82Formula: (2*3 - 1)2^{4} + 2 = (6 - 1)*16 + 2 = 5*16 + 2 = 80 + 2 = 82. Correct.n=4:S_4 = 3*2 + 5*4 + 7*8 + 9*16 = 6 + 20 + 56 + 144 = 226Formula: (2*4 - 1)2^{5} + 2 = (8 - 1)*32 + 2 = 7*32 + 2 = 224 + 2 = 226. Correct.Okay, so the formula seems to work for these small values. So, I think I did it correctly.Therefore, the sum S_n is (2n - 1)2^{n+1} + 2.Alternatively, I can write this as 2^{n+1}(2n - 1) + 2.But perhaps it can be simplified further or written in another form. Let me see.Alternatively, factor 2^{n+1}:= 2^{n+1}(2n - 1) + 2Alternatively, factor 2:= 2( (2n - 1)2^n + 1 )But I think the first form is fine.So, summarizing:(1) The general formula for the arithmetic sequence is a_n = 2n + 1.(2) The sum S_n = (2n - 1)2^{n+1} + 2.I think that's the answer.**Final Answer**(1) The general formula for the sequence is boxed{a_n = 2n + 1}.(2) The sum of the first ( n ) terms of the sequence ({a_n cdot 2^n}) is boxed{S_n = 2 - (1 - 2n) times 2^{n+1}}.Wait, hold on. In my earlier calculation, I got S_n = (2n - 1)2^{n+1} + 2. But the final answer given in the initial problem was S_n = 2 - (1 - 2n)2^{n+1}. Let me check if these are equivalent.Let me expand 2 - (1 - 2n)2^{n+1}:= 2 - 2^{n+1} + 2n*2^{n+1}= 2 + (2n - 1)2^{n+1}Which is the same as what I had: (2n - 1)2^{n+1} + 2. So, yes, they are the same. So, both forms are correct, just written differently.Therefore, the answer can be written as 2 - (1 - 2n)2^{n+1}, which is the same as (2n - 1)2^{n+1} + 2.So, to match the initial problem's answer, I can write it as 2 - (1 - 2n)2^{n+1}.Thus, the final answers are:(1) boxed{a_n = 2n + 1}(2) boxed{S_n = 2 - (1 - 2n) times 2^{n+1}}