2,17,82,257,626,1297 next one please ?

The easy thing to do is notice that 1^4 = 1, 2^4 = 16, 3^4 = 81, and so on, so the sequence follows the rule . The next number would then be fourth power of 7 plus 1, or 2402.

And the harder way: Denote the n-th term in this sequence by , and denote the given sequence by .

Let denote the n-th term in the sequence of forward differences of , defined by

for n ≥ 1. That is, is the sequence with

and so on.

Next, let denote the n-th term of the differences of , i.e. for n ≥ 1,

so that

etc.

Again: let denote the n-th difference of :

etc.

One more time: let denote the n-th difference of :

etc.

The fact that these last differences are constant is a good sign that for all n ≥ 1. Assuming this, we would see that is an arithmetic sequence given recursively by

and we can easily find the explicit rule:

and so on, up to

Use the same strategy to find a closed form for , then for , and finally .

and so on, up to

Recall the formula for the sum of consecutive integers:

and so on, up to

Recall the formula for the sum of squares of consecutive integers:

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