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Carlos and Ana were asked to find an explicit formula for the sequence -10,-4,2,8,dots, where the first term should be f(1). Carlos said the formula is f(n)=-10+6n. Ana said the formula is f(n)=-10+6(n-1). Which one of them is right? Choose 1 answer: (A) Only Carlos (B) Only Ana (c) Both Carlos and Ana (D) Neither Carlos nor Ana

Carlos and Ana were asked to find an explicit formula for the sequence −10,−4,2,8,… -10,-4,2,8, \ldots , where the first term should be f(1) f(1) .\newlineCarlos said the formula is f(n)=−10+6n f(n)=-10+6 n .\newlineAna said the formula is f(n)=−10+6(n−1) f(n)=-10+6(n-1) .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Carlos\newline(B) Only Ana\newline(C) Both Carlos and Ana\newline(D) Neither Carlos nor Ana

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Q. Carlos and Ana were asked to find an explicit formula for the sequence −10,−4,2,8,… -10,-4,2,8, \ldots , where the first term should be f(1) f(1) .\newlineCarlos said the formula is f(n)=−10+6n f(n)=-10+6 n .\newlineAna said the formula is f(n)=−10+6(n−1) f(n)=-10+6(n-1) .\newlineWhich one of them is right?\newlineChoose 11 answer:\newline(A) Only Carlos\newline(B) Only Ana\newline(C) Both Carlos and Ana\newline(D) Neither Carlos nor Ana
  1. Identify sequence type and pattern: Identify the type of sequence and the pattern.\newlineThe sequence given is −10,−4,2,8,…-10, -4, 2, 8, \ldots which increases by 66 each time. This indicates that it is an arithmetic sequence with a common difference (dd) of 66.
  2. Write explicit formula for arithmetic sequence: Write the general form of the explicit formula for an arithmetic sequence.\newlineThe general form of the explicit formula for an arithmetic sequence is f(n)=a1+(n−1)df(n) = a_1 + (n-1)d, where a1a_1 is the first term and dd is the common difference.
  3. Determine first term of sequence: Determine the first term (a1a_1) of the sequence.\newlineThe first term given in the sequence is −10-10, so a1=−10a_1 = -10.
  4. Substitute values into formula: Substitute the values of a1a_1 and dd into the general formula.\newlineSubstituting a1=−10a_1 = -10 and d=6d = 6 into the formula gives us f(n)=−10+(n−1)6f(n) = -10 + (n-1)6.
  5. Simplify the formula: Simplify the formula.\newlineSimplifying the formula, we get f(n)=−10+6n−6f(n) = -10 + 6n - 6, which further simplifies to f(n)=6n−16f(n) = 6n - 16.
  6. Compare formulas with Carlos and Ana: Compare the simplified formula to the formulas given by Carlos and Ana.\newlineCarlos's formula is f(n)=−10+6nf(n) = -10 + 6n, which is not the same as our simplified formula.\newlineAna's formula is f(n)=−10+6(n−1)f(n) = -10 + 6(n-1), which simplifies to f(n)=6n−16f(n) = 6n - 16, matching our simplified formula.
  7. Determine correct formula: Determine which person is correct.\newlineSince Ana's formula matches the simplified formula we derived, Ana is correct. Carlos's formula does not match, so he is incorrect.

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