3 + 7 + 14 + 24 + 37 + …

Solution:

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

We have,

S_n = 3 + 7 + 14 + 24 + 37 + …………. + T_n-1 + T_n … (1)

Equation (1) can be rewritten as:

S_n = 3 + 7 + 14 + 24 + 37 + …………. + T_n-1 + T_n ……..(2)

By subtracting (2) from (1) we get

S_n = 3 + 7 + 14 + 24 + 37 + …………. + T_n-1 + T_n

S_n = 3 + 7 + 14 + 24 + 37 + …………. + T_n-1 + T_n

0 = 3 + [4 + 7 + 10 + 13 + … + (T_n – T_n-1)] – T_n

The difference between the successive terms are 4, 7, 10, 13

So these differences are in A.P

Now,

∴ The sum of the series is n/2 [n² + n + 4]