Question

How do you find the explicit formula and calculate term 20 for -1, 6, 25, 62, 123?

Answer 1

Look at sequences of differences to construct a formula and find the `20`th term is `7998`.

Explanation:

Each of the last 3 terms looks close to a cube, so I could guess the formula as `a_n = n^3-2`, but let's pretend I didn't spot that...

Write out the initial sequence:

`color(blue)(-1), 6, 25, 62, 123`

Write out the sequence of differences of that sequence:

`color(blue)(7), 19, 37, 61`

Write out the sequence of differences of that sequence:

`color(blue)(12), 18, 24`

Write out the sequence of differences of that sequence:

`color(blue)(6), 6`

Having reached a constant sequence, we can now use the first number of each sequence to write out a formula for `a_n` as follows:

`a_n = color(blue)(-1)/(0!) +color(blue)(7)/(1!) (n-1) + color(blue)(12)/(2!) (n-1)(n-2) + color(blue)(6)/(3!) (n-1)(n-2)(n-3)`

`=-1+7(n-1)+6(n-1)(n-2)+(n-1)(n-2)(n-3)`

`=-1+7n-7+6n^2-18n+12+n^3-6n^2+11n-6`

`=n^3-2`

So `a_20 = 20^3-2 = 8000-2 = 7998`