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Gustavo and Yuki were asked to find an explicit formula for the sequence 19,0,-19,-38,dots, where the first term should be h(1). Gustavo said the formula is h(n)=19-19 n. Yuki said the formula is h(n)=-19+19 n. Which one of them is right? Choose 1 answer: A Only Gustavo (B) Only Yuki (C) Both Gustavo and Yuki (D) Neither Gustavo nor Yuki

Gustavo and Yuki were asked to find an explicit formula for the sequence 19,0,−19,−38,… 19,0,-19,-38, \ldots , where the first term should be h(1) h(1) .\newlineGustavo said the formula is h(n)=19−19n h(n)=19-19 n .\newlineYuki said the formula is h(n)=−19+19n h(n)=-19+19 n .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Gustavo\newline(B) Only Yuki\newline(C) Both Gustavo and Yuki\newline(D) Neither Gustavo nor Yuki

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Q. Gustavo and Yuki were asked to find an explicit formula for the sequence 19,0,−19,−38,… 19,0,-19,-38, \ldots , where the first term should be h(1) h(1) .\newlineGustavo said the formula is h(n)=19−19n h(n)=19-19 n .\newlineYuki said the formula is h(n)=−19+19n h(n)=-19+19 n .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Gustavo\newline(B) Only Yuki\newline(C) Both Gustavo and Yuki\newline(D) Neither Gustavo nor Yuki
  1. Identify pattern in sequence: Identify the pattern in the sequence. The sequence 1919, 00, −19-19, −38-38, ... decreases by 1919 each time. This indicates that it is an arithmetic sequence with a common difference of −19-19.
  2. Determine first term: Determine the first term of the sequence. The first term, h(1)h(1), is 1919.
  3. Use explicit formula: Use the explicit formula for an arithmetic sequence, h(n)=h(1)+(n−1)dh(n) = h(1) + (n-1)d, where h(1)h(1) is the first term and dd is the common difference. For this sequence, h(1)=19h(1) = 19 and d=−19d = -19.
  4. Substitute values into formula: Substitute the values of h(1)h(1) and dd into the formula to find the explicit formula for the sequence. The formula becomes h(n)=19+(n−1)(−19)h(n) = 19 + (n-1)(-19).
  5. Simplify the formula: Simplify the formula. h(n)=19−19(n−1)=19−19n+19=19(1−n)=19−19nh(n) = 19 - 19(n-1) = 19 - 19n + 19 = 19(1 - n) = 19 - 19n.
  6. Compare with Gustavo and Yuki: Compare the simplified formula to the formulas provided by Gustavo and Yuki. Gustavo's formula is h(n)=19−19nh(n) = 19 - 19n, which matches our derived formula. Yuki's formula is h(n)=−19+19nh(n) = -19 + 19n, which does not match our derived formula.

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