Question

How do you find the nth term of the sequence `1, 3, 6, 10, 15,...`?

Answer 1

`a_n = 1/2n(n+1)`

Explanation:

These are recognisable as triangular numbers, but let's use a general method for finding matching polynomial formulas...

Write down the initial sequence:

`color(red)(1), 3, 6, 10, 15`

Write down the sequence of differences between consecutive pairs of terms:

`color(magenta)(2), 3, 4, 5`

Write down the sequence of differences of those differences:

`color(blue)(1), 1, 1`

Having reached a constant sequence, we can write down a formula for the `n`th term using the initial term of each of these sequences as a coefficient:

`a_n = color(red)(1)/(0!) + color(magenta)(2)/(1!)(n-1) + color(blue)(1)/(2!)(n-1)(n-2)`

`color(white)(a_n) = 1+2n-2+1/2n^2-3/2n+1`

`color(white)(a_n) = 1/2n^2+1/2n`

`color(white)(a_n) = 1/2n(n+1)`