Question

If mc025-1.jpg, which recursive formula could represent the sequence below?

..., 24, 88, 664, 8408, ...
mc025-2.jpg
mc025-3.jpg
mc025-4.jpg
mc025-5.jpg

Answer 1

Look at the differences:
88 − 24 = 64 = 8^2
664 − 88 = 576 = 24^2
8408 − 664 = 7744 = 88^2

So each term equals the previous term plus the square of the term before that. A convenient recursive description is

a1 = 8, a2 = 24, and for n ≥ 3
a_n = a_{n-1} + (a_{n-2})^2

(or equivalently a_{n+1} = a_n + a_{n-1}^2).