Question

Sum the series to n term:

1 + 3 + 6 + 10 + 15 + …

Answer 1

Let T_n be the n^th term and S_n be the sum to n terms of the given series.

We have,

S_n = 1 + 3 + 6 + 10 + 15 + …………. + T_n-1 + T_n … (1)

Equation (1) can be rewritten as:

S_n = 1 + 3 + 6 + 10 + 15 + …………. + T_n-1 + T_n ……..(2)

By subtracting (2) from (1) we get

S_n = 1 + 3 + 6 + 10 + 15 + …………. + T_n-1 + T_n

S_n = 1 + 3 + 6 + 10 + 15 + …………. + T_n-1 + T_n

0 = 1 + [2 + 3 + 4 + 5 + … + (T_n – T_n-1)] – T_n

The difference between the successive terms are 2, 3, 4, 5

So these differences are in A.P

Now,

∴ The sum of the series is n/6 (n + 1) (n + 2)