Question

The sum of first 20 terms of an A.P. is equal to the sum of the first 30 terms. Show that the sum of the first 50 terms of an A.P. is zero

Answer 1

This is the solutiom to your prpblem

Answer 2

We know that the sum of 'n' terms of an A.P is given by :

Sn = n/2 × { 2a + (n - 1)d }

⇒ Sum of first 20 terms = 20/2 × { 2a + (20 - 1)d }

⇒ Sum of first 20 terms = 10(2a + 19d) = 20a + 190d

⇒ Sum of first 30 terms = 30/2 × { 2a + (30 - 1)d }

⇒ Sum of first 30 terms = 15 {2a + 29d} = 30a + 435d

Given that Sum of 1st 20 terms = Sum of 1st 30 terms

⇒ 20a + 190d = 30a + 435d

⇒ 10a + 245d = 0

⇒ 2a + 49d = 0

Sum of first 50 terms = 50/2 × { 2a + (50 - 1)d }

⇒ Sum of first 50 terms  = 25(2a + 49d)

But we found that (2a + 49d) = 0

⇒ Sum of first 50 terms = 0