The nth term of the sequence $a_{1}$ , $a_{2}$ , $a_{3}$ ,..., is given by the equation $a_{n}$ = $(a_{n–1})^{2}$ + n. If $a_{3}$ – $(a_{1})^{4}$ = 31, what is $a_{2}$ ?

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Breaking Down the Info:

We have

a_3 = (a_2)^2 + 3 = ((a_1)^2 + 2)^2 + 3 = (a_1)^4 + 4*(a_1)^2 + 4 + 3

.

Then we can plug in to find

4*(a_1)^2 + 4 + 3 = 31

and

(a_1)^2 = 6

.…

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What this question tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.