For the sequence of 15 16 19 20 23, what is the nth term and what is the formula for finding it?

Hello!

Consider the pairwise differences between the terms of this sequence. They are 16 - 15 = 1, 19 - 16 = 3, 20 - 19 =1, 23 - 20 = 3.

So we can suppose that the next difference will be 1, then 3, then 1 again, then 3...

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Hello!

Consider the pairwise differences between the terms of this sequence. They are 16 - 15 = 1, 19 - 16 = 3, 20 - 19 =1, 23 - 20 = 3.

So we can suppose that the next difference will be 1, then 3, then 1 again, then 3 again and so on.

It is simple to find separate formulas for odd and even n's. They are clearly

`a_n =13+2n,` n is odd, `a_n = 12 + 2n,` n is even.

Actually, it is a legitimate formula (or at least algorithm) to find any term of the sequence. If we want to get "one formula," we can use `(-1)^n,` or even `(1+(-1)^n)/2, ` which gives us the sequence 0, 1, 0, 1 and so on.

With such an addition the formula becomes

`a_n = 12 + ((1+(-1)^(n+1))/2)+2n.`

The next term is `a_6 = 12+0+12=24.`

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