Question

A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying Sk+1 completely. Show your work. Sn: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3

Answer 1

Alright, for the sum part, it really seems like it's x*(x+1) +(x+1)*(x+2)repeating forever. Therefore, it is ∑ (1 at the bottom, n at the top) k*(k+1)=[n(n + 1)(n + 2)]/3

Answer 2

Answer with explanation:

`S_(n)= 1 * 2 + 2 * 3 + 3 * 4 + . . . + n*(n + 1) = (n(n + 1)(n + 2))/(3)\\\\S_(k)= 1 * 2 + 2 * 3 + 3 * 4 + . . . + k*(k + 1) = (k *(k + 1)(k + 2))/(3)\\\\S_(k+1)= 1 * 2 + 2 * 3 + 3 * 4 + . . . + k*(k + 1)+(k+1) * (k+2)=((k+1) *(k +1+ 1)(k +1+ 2))/(3)\\\\=((k+1) *(k +2)*(k +3))/(3)`