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

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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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