Question

If the sum of first 11 terms of an arithmetic progression equal that of the first 19 terms then what is the sum of first 30 terms???​

Answer 1

SOLUTION:-

Given:

If the sum of first 11th term of an A.P. equal that of first 19 terms.

Therefore,

We know that, Formula of the sum of A.P.

${}^{S} n = \frac{n}{2} [2a + (n - 1)d]$

So,

${}^{S} 11 = \frac{11}{2} [2a + (11 - 1)d]\\ \\ = > 11(a + 5d)..............(1) \\ \\ {}^{S} 19 = \frac{19}{2} (2a +(19 - 1)d) \\ \\ = > 19(a + 9)d...............(2)$

Comparing equation (1) & (2), we get;

${}^{S} 11 = {}^{s} 19 \\ \\ = > 11(a + 5d) = 19(a + 9d) \\ \\ = > 11a + 55d = 19a + 171d \\ \\ = > 19a - 11a= 171d - 55d \\ \\ = > 8a + 116d = 0\\ \\ = > 2a + 29d = 0$

Now,

30th term;

${}^{S} 30 = \frac{30}{2} ( 2a + 29d) \\ \\ = > 0$

Hope it helps ☺️