The sum of the first five terms of an AP and the sum of the first seven terms of the same AP is 167. If the sum of the first ten terms of this AP is 235, find the sum of its first twenty terms.

Given: S₅ + S₇ = 167

S₁₀ = 235

To find: S₂₀

Formula Used:

Sum of “n” terms of an AP:

Where S_n is the sum of first n terms

n = no of terms

a = first term

d = common difference

Explanation:

Let the first term, common difference and the number of terms of an AP are a, d and n, respectively.

As,

S₅ + S₇ = 167

we have,

5(2a + 4d) + 7( 2a + 6d ) = 334

10a + 20d + 14a + 42d = 334

24a + 62d = 334

12a + 31d = 167

12a = 167 - 31d [ eqn 1]

Also,

S₁₀ = 235

5[ 2a + 9d] = 235

2a + 9d = 47

12a + 54d = 282 [ multiplication by 6 both side]

167 - 31d + 54d = 282 [ using equation 1]

23d = 282 - 167

23d = 115

d = 5

using this value in equation 1

12a = 167 - 31(5 )

12a = 167 - 155

12a = 12

a = 1

Now

= 10[ 2 + 95]

= 970

So, the sum of first 20 terms is 970.