Question

Is the sequence "`4, 16, 36, 64,...`" arithmetic?

Answer 1

No, it has no common difference between successive terms.

The terms are given by the formula `a_n = 4 n^2` where `n = 1,2,3,...`

Explanation:

Write out the original sequence:

`color(blue)(4), 16, 36, 64`

Write out the sequence of differences of the sequence:

`color(blue)(12), 20, 28`

Since this is not constant, the sequence is not arithmetic.

Write out the sequence of differences of that sequence:

`color(blue)(8), 8`

This is a constant sequence, so we can derive a polynomial formula for the terms of the sequence from the initial term of each of these sequences:

`a_n = color(blue)(4)/(0!) + color(blue)(12)/(1!)(n-1) + color(blue)(8)/(2!)(n-1)(n-2)`

`= 4+12n-12+4n^2-12n+8 = 4n^2`