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:

there are 70 more legs than eyes at the dog show (196 – 126 = 70). dogs have two more legs than eyes, and humans have the same number of legs as eyes. thus 70 is twice the number of dogs at the show, so the number of dogs is half of 70, which is 35. to get the number of humans, subtract 35 from 63: 63 – 35 = 28.

step-by-step explanation:

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