Question

What is the value of the sum of the sequence: `1+5+14+30+...........` upto n terms? Thank you!

Answer 1

`s_n = 1/12(n^4 + 4n^3 + 5n^2 + 2n)`

Explanation:

Notice that the sequence of differences between terms follows the pattern `4`, `9`, `16`, i.e. `2^2`, `3^2`, `4^2`.

Assuming the pattern continues, the next couple of terms would be `55` and `91`, being `30+5^2` and `55+6^2`

Write down the sequence of the first few sums:

`color(blue)(1), 6, 20, 50, 105, 196`

Write down the sequence of differences between consecutive terms:

`color(blue)(5), 14, 30, 55, 91`

Write down the sequence of differences of that sequence:

`color(blue)(9), 16, 25, 36`

Write down the sequence of differences of that sequence:

`color(blue)(7), 9, 11`

Write down the sequence of differences of that sequence:

`color(blue)(2), 2`

Having reached a constant sequence, we can use the initial terms of each of these sequences as coefficients of an expression for the `n`th term of the sequence of sums:

`s_n = color(blue)(1)/(0!) + color(blue)(5)/(1!)(n-1) + color(blue)(9)/(2!)(n-1)(n-2) + color(blue)(7)/(3!)(n-1)(n-2)(n-3) + color(blue)(2)/(4!)(n-1)(n-2)(n-3)(n-4)`

`color(white)(s_n) = 1+5(n-1)+9/2(n^2-3n+2)+7/6(n^3-6n^2+11n-6)+1/12(n^4 - 10n^3 + 35n^2 - 50n + 24)`

`color(white)(s_n) = 1/12(n^4 + 4n^3 + 5n^2 + 2n)`