Question

Which is the recursive formula for the sequence whose explicit formula is an = (n + 1)2?

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Answer 1

Do you mean a_n = (n+1)^2 (square) or a_n = 2(n+1) (times 2)? The recursive formulas are different:

- If a_n = (n+1)^2 (n starting at 0): a_0 = 1 and for n ≥ 1
a_n = a_{n-1} + (2n + 1).
(Because (n+1)^2 − n^2 = 2n + 1.)

- If a_n = 2(n+1) = 2n+2 (n starting at 0): a_0 = 2 and for n ≥ 1
a_n = a_{n-1} + 2.

If you paste the four choices (or confirm which interpretation), I’ll pick the matching one.