💡Alright, so I've got this math problem here with five statements about opposite numbers, and I need to figure out which ones are correct. Let me take it step by step.First, let's make sure I understand what opposite numbers are. From what I remember, opposite numbers are numbers that are the same distance from zero on the number line but in opposite directions. So, for example, the opposite of 5 is -5, and the opposite of -3 is 3. That makes sense because they're both three units away from zero but on opposite sides.Okay, let's look at the first statement: "The opposite of π is -π." Hmm, π is approximately 3.14159, right? So, if I take the opposite of π, that should be -3.14159, which is indeed -π. That seems straightforward. So, statement ① is correct.Next, statement ② says, "Numbers with opposite signs are opposite numbers to each other." Wait a minute, does that mean any two numbers with different signs are opposites? For example, is 5 and -3 considered opposites? Well, according to the definition I just recalled, opposites need to have the same magnitude but opposite signs. So, 5 and -5 are opposites, but 5 and -3 are not because their magnitudes are different. Therefore, statement ② is not entirely correct because it doesn't specify that the numbers need to have the same magnitude.Moving on to statement ③: "The opposite of -3.8 is 3.8." Let's see, if I take the opposite of -3.8, I just change its sign, right? So, -(-3.8) equals 3.8. That checks out. So, statement ③ is correct.Statement ④ says, "A number and its opposite may be equal." Hmm, when would a number be equal to its opposite? Let's think about this. If I have a number x and its opposite is -x, when would x equal -x? The only number that satisfies this is 0 because -0 is still 0. So, in this case, 0 is equal to its opposite. Therefore, statement ④ is correct.Finally, statement ⑤: "Positive numbers and negative numbers are opposite to each other." This sounds similar to statement ②. Are all positive and negative numbers opposites? No, because for them to be opposites, they need to have the same magnitude. For example, 2 and -3 are both positive and negative numbers, but they're not opposites because their magnitudes are different. So, statement ⑤ is incorrect.Let me recap:- Statement ①: Correct- Statement ②: Incorrect- Statement ③: Correct- Statement ④: Correct- Statement ⑤: IncorrectSo, out of the five statements, three are correct: ①, ③, and ④. That means the correct option is D: 3.

💡First, I need to determine the man's regular pay for 40 hours by multiplying his regular hourly rate by the number of regular hours.Next, I'll calculate the total pay he received and subtract the regular pay to find out how much of his pay was for overtime hours.Since overtime pay is twice the regular pay rate, I'll multiply the regular hourly rate by 2 to find the overtime rate.Finally, I'll divide the overtime pay by the overtime rate to determine the number of overtime hours he worked.

💡Okay, so I have this problem here: Given the constant ( a in mathbb{R} ), if the coefficient of the ( x^7 ) term in the binomial expansion of ( left( x^2 + frac{a}{x} right)^5 ) is ( -15 ), find the value of ( a ).Hmm, binomial expansion. I remember that the binomial theorem allows us to expand expressions of the form ( (A + B)^n ). The general term in such an expansion is given by ( T_{k+1} = binom{n}{k} A^{n-k} B^k ), right? So, in this case, ( A ) is ( x^2 ) and ( B ) is ( frac{a}{x} ), and ( n ) is 5.So, the general term in the expansion of ( left( x^2 + frac{a}{x} right)^5 ) would be ( binom{5}{k} (x^2)^{5 - k} left( frac{a}{x} right)^k ). Let me write that out:( T_{k+1} = binom{5}{k} (x^2)^{5 - k} left( frac{a}{x} right)^k )Simplifying this, I can combine the exponents of ( x ). Let's see:( (x^2)^{5 - k} = x^{2(5 - k)} = x^{10 - 2k} )And ( left( frac{a}{x} right)^k = a^k cdot x^{-k} )So, multiplying these together, the term becomes:( binom{5}{k} a^k x^{10 - 2k - k} = binom{5}{k} a^k x^{10 - 3k} )Okay, so each term has an exponent of ( x ) equal to ( 10 - 3k ). We need the term where the exponent is 7, so I can set up the equation:( 10 - 3k = 7 )Solving for ( k ):( 10 - 7 = 3k )( 3 = 3k )( k = 1 )So, the term corresponding to ( k = 1 ) will have the exponent ( x^7 ). Let's find the coefficient for this term.The coefficient is ( binom{5}{1} a^1 ). Calculating the binomial coefficient:( binom{5}{1} = 5 )So, the coefficient is ( 5a ). According to the problem, this coefficient is ( -15 ). Therefore:( 5a = -15 )Solving for ( a ):( a = frac{-15}{5} = -3 )So, the value of ( a ) is ( -3 ).Wait, let me double-check to make sure I didn't make a mistake. So, the general term is ( binom{5}{k} a^k x^{10 - 3k} ). For ( x^7 ), ( 10 - 3k = 7 ) gives ( k = 1 ). Then, the coefficient is ( binom{5}{1} a = 5a ). Setting that equal to ( -15 ), so ( 5a = -15 ) leads to ( a = -3 ). Yeah, that seems correct.I think I've got it. The value of ( a ) is ( -3 ).**Final Answer**The value of ( a ) is boxed{-3}.

💡To determine the least possible value of ( y ), let's analyze the given conditions step by step.1. **Identify the smallest even integer for ( x ):** - The smallest even integer is ( 0 ).2. **Determine the smallest odd integer ( z ) such that ( z - x = 9 ):** - If ( x = 0 ), then ( z = 9 ) to satisfy ( z - x = 9 ).3. **Find the smallest odd integer ( y ) such that ( y - x > 5 ):** - With ( x = 0 ), ( y ) must be greater than ( 5 ). - The smallest odd integer greater than ( 5 ) is ( 7 ).Therefore, the least possible value of ( y ) is ( 7 ).

💡Alright, so I have this sequence problem here. It says that in the sequence {a_n}, a₁ is 1, and the difference between consecutive terms, a_{n+1} - a_n, is equal to sin((n+1)π/2). I need to find S_{2017}, which is the sum of the first 2017 terms of this sequence. Hmm, okay, let's break this down step by step.First, I know that a₁ is given as 1. So, a₁ = 1. Now, to find a₂, I can use the recurrence relation: a_{n+1} - a_n = sin((n+1)π/2). So, for n=1, a₂ - a₁ = sin(2π/2) = sin(π). I remember that sin(π) is 0, so a₂ - a₁ = 0, which means a₂ = a₁ = 1.Okay, moving on to a₃. For n=2, a₃ - a₂ = sin(3π/2). I recall that sin(3π/2) is -1, so a₃ - a₂ = -1. Since a₂ is 1, that means a₃ = 1 - 1 = 0.Next, let's find a₄. For n=3, a₄ - a₃ = sin(4π/2) = sin(2π). Sin(2π) is 0, so a₄ - a₃ = 0, meaning a₄ = a₃ = 0.Now, let's compute a₅. For n=4, a₅ - a₄ = sin(5π/2). Sin(5π/2) is sin(π/2 + 2π) which is the same as sin(π/2) because sine has a period of 2π. Sin(π/2) is 1, so a₅ - a₄ = 1. Since a₄ is 0, a₅ = 0 + 1 = 1.Wait a minute, a₅ is 1, which is the same as a₁. Let me check the next term to see if this is part of a repeating pattern. For n=5, a₆ - a₅ = sin(6π/2) = sin(3π). Sin(3π) is 0, so a₆ - a₅ = 0, meaning a₆ = a₅ = 1.Hmm, a₆ is also 1. Let's go to a₇. For n=6, a₇ - a₆ = sin(7π/2). Sin(7π/2) is sin(3π + π/2) which is the same as sin(π/2) because sine has a period of 2π, but wait, sin(7π/2) is actually sin(3π + π/2) which is sin(π/2) but with a sign change because it's in the third quadrant. So, sin(7π/2) is -1. Therefore, a₇ - a₆ = -1, so a₇ = 1 - 1 = 0.Continuing, for n=7, a₈ - a₇ = sin(8π/2) = sin(4π) = 0. So, a₈ = a₇ = 0.And for n=8, a₉ - a₈ = sin(9π/2). Sin(9π/2) is sin(π/2 + 4π) which is sin(π/2) = 1. So, a₉ = a₈ + 1 = 0 + 1 = 1.Wait, so a₉ is 1 again. It seems like the sequence is repeating every 4 terms: 1, 1, 0, 0, 1, 1, 0, 0, and so on. Let me confirm this pattern.So, from a₁ to a₄: 1, 1, 0, 0.a₅ to a₈: 1, 1, 0, 0.a₉ to a₁₂: 1, 1, 0, 0.Yes, it does seem to repeat every 4 terms. So, the sequence {a_n} is periodic with period 4. That is, a_{n+4} = a_n for all n.Now, since the sequence repeats every 4 terms, the sum of each block of 4 terms is the same. Let's compute the sum of one period: a₁ + a₂ + a₃ + a₄ = 1 + 1 + 0 + 0 = 2.So, each block of 4 terms adds up to 2. Now, to find S_{2017}, which is the sum of the first 2017 terms, we can figure out how many complete blocks of 4 are in 2017 terms and then add the remaining terms.First, let's divide 2017 by 4 to find out how many complete blocks there are and the remainder.2017 divided by 4 is 504 with a remainder of 1, because 4*504 = 2016, and 2017 - 2016 = 1.So, there are 504 complete blocks of 4 terms, each summing to 2, and then 1 additional term.Therefore, the total sum S_{2017} is 504*2 + a₁, since the remainder is 1, which is the first term of the next block.But wait, let me make sure. The first term after 2016 terms is a_{2017}, which is the same as a₁ because the sequence repeats every 4 terms. So, a_{2017} = a₁ = 1.Therefore, S_{2017} = 504*2 + 1 = 1008 + 1 = 1009.Wait, let me double-check my calculations. 504 blocks of 4 terms: 504*4 = 2016 terms, so the sum is 504*2 = 1008. Then, adding the 2017th term, which is a₁ = 1, so total sum is 1008 + 1 = 1009.Yes, that seems correct. So, S_{2017} is 1009.

💡First, I need to determine the amount by which the price of the lawn chair decreased. This can be found by subtracting the sale price from the original price.Next, I'll calculate the percentage decrease by dividing the price difference by the original price and then multiplying by 100 to convert it into a percentage.Finally, I'll round the result to the nearest whole number to provide an approximate percentage decrease.