Sum of Arithmetic series

Question

Sum of consecutive values can be found easily. But I can't figure it out how to find the closed form of the following arithmetic series? Can anybody explain it elaborately?

$ S = (1) + (1+2) + (1+2+3) + (1+2+3+4) + \dots + (1+2+3+\dots+n) $.

Thanks in Advance.

See also: http://math.stackexchange.com/questions/376284/sum-of-the-first-n-triangular-numbers-induction — Martin Sleziak, Oct 08 '15 at 01:14

score 2 · Accepted Answer · edited Apr 04 '15 at 23:10

2

Hints

We have $1+2+\cdots+n=\frac{n(n+1)}{2}$
And your $S=\frac{1(1+1)}{2}+\frac{2(2+1)}{2}+\cdots+\frac{n(n+1)}{2}$ you can prove by induction :$$S=\frac{n(n+1)(n+2)}{6}$$

edited Apr 04 '15 at 23:10

Cookie

13,532

answered Apr 04 '15 at 23:09

Elaqqad

13,725

Sum of Arithmetic series

1 Answers1