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.

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


Given S5 + S7 = 167


Using the formula,


Where Sn is the sum of first n terms


So 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.


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