Question

A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work. (refer to image)?

Answer 1

`S_k = (k(k+1)(k+2))/3`
`S_(k+1) = ((k+1)(k+2)(k+3))/3`

Explanation:

We are given that `S_n = (n(n+1)(n+2))/3`.

To find `S_k`, simply replace all occurrences of `n` in `S_n` with `k`. That is, `S_k = (k(k+1)(k+2))/3`.

To find `S_(k+1)`, replace all instances of `n` in `S_n` with `k+1`. This gives `S_(k+1) = ((k+1)((k+1)+1)((k+1)+2))/3`. Combining terms, this gives `S_(k+1) = ((k+1)(k+2)(k+3))/3`.