Question

What is the next term in the sequence `1, 9, 29, 67,...` ?

Answer 1

It could be `129`

Explanation:

Given:

`1, 9, 29, 67`

Compare with the first few cubes:

`1, 8, 27, 64`

The differences are:

`0, 1, 2, 3`

So at least one matching formula is:

`a_n = n^3+n-1`

Then:

`x = a_5 = (color(blue)(5))^3+(color(blue)(5))-1 = 125+5-1 = 129`

Note that rather than spotting this directly, we could use a method of differences to find any polynomial sequence.

In our example, write down the initial sequence:

`color(blue)(1), 9, 29, 67`

Write down the sequence of differences between consecutive terms:

`color(blue)(8), 20, 38`

Write down the sequence of differences of those differences:

`color(blue)(12), 18`

Write down the sequence of differences of those differences:

`color(blue)(6)`

Having reached a constant sequence (albeit of just one term), we can write down a formula for the general term using the first term of each of these sequences as coefficients:

`a_n = color(blue)(1)/(0!) + color(blue)(8)/(1!)(n-1) + color(blue)(12)/(2!)(n-1)(n-2) + color(blue)(6)/(3!)(n-1)(n-2)(n-3)`

`color(white)(a_n) = 1+8n-8+6n^2-18n+12+n^3-6n^2+11n-6`

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