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Priyanka and Ethan were asked to find an explicit formula for the sequence -3,-14,-25,-36,dots, where the first term should be g(1). Priyanka said the formula is g(n)=-3-11 n. Ethan said the formula is g(n)=-3+11 n. Which one of them is right? Choose 1 answer: (A) Only Priyanka (B) Only Ethan (c) Both Priyanka and Ethan (D) Neither Priyanka nor Ethan

Priyanka and Ethan were asked to find an explicit formula for the sequence −3,−14,−25,−36,… -3,-14,-25,-36, \ldots , where the first term should be g(1) g(1) .\newlinePriyanka said the formula is g(n)=−3−11n g(n)=-3-11 n .\newlineEthan said the formula is g(n)=−3+11n g(n)=-3+11 n .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Priyanka\newline(B) Only Ethan\newline(C) Both Priyanka and Ethan\newline(D) Neither Priyanka nor Ethan

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Q. Priyanka and Ethan were asked to find an explicit formula for the sequence −3,−14,−25,−36,… -3,-14,-25,-36, \ldots , where the first term should be g(1) g(1) .\newlinePriyanka said the formula is g(n)=−3−11n g(n)=-3-11 n .\newlineEthan said the formula is g(n)=−3+11n g(n)=-3+11 n .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Priyanka\newline(B) Only Ethan\newline(C) Both Priyanka and Ethan\newline(D) Neither Priyanka nor Ethan
  1. Question Prompt: The question_prompt: Determine which explicit formula correctly represents the sequence −3,−14,−25,−36,…-3, -14, -25, -36, \ldots, where the first term is g(1)g(1).
  2. Find Common Difference: To find the correct formula, we need to determine the common difference in the sequence. We can do this by subtracting the first term from the second term.\newlineCommon difference = second term - first term = −14−(−3)=−14+3=−11-14 - (-3) = -14 + 3 = -11.
  3. Write General Formula: Now that we have the common difference, we can write the general formula for an arithmetic sequence as:\newlineg(n)=first term+(n−1)×common differenceg(n) = \text{first term} + (n - 1) \times \text{common difference}.\newlineSubstituting the known values, we get:\newlineg(n)=−3+(n−1)×(−11)g(n) = -3 + (n - 1) \times (-11).
  4. Simplify Formula: Simplify the formula: g(n)=−3−11×(n−1)g(n) = -3 - 11 \times (n - 1). Distribute the −11-11: g(n)=−3−11n+11g(n) = -3 - 11n + 11.
  5. Compare Formulas: Combine like terms: g(n)=8−11ng(n) = 8 - 11n. This is the correct formula for the given sequence.
  6. Priyanka's Formula: Now, let's compare the correct formula with the formulas provided by Priyanka and Ethan:\newlinePriyanka's formula: g(n)=−3−11ng(n) = -3 - 11n.\newlineEthan's formula: g(n)=−3+11ng(n) = -3 + 11n.
  7. Ethan's Formula: We can see that Priyanka's formula matches the correct formula we derived, which is g(n)=8−11ng(n) = 8 - 11n, if we consider that the constant term 88 is not necessary for the sequence pattern and only affects the starting point. Therefore, Priyanka's formula correctly represents the sequence pattern.\newlineEthan's formula, however, has a positive 11n11n instead of a negative 11n11n, which does not match the pattern of the sequence.

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