In an AP, if S5 + S7 = 167 and S10 = 235, then find the AP, where Sn denotes the sum of its first n terms.

Question

Philoid · Accepted Answer

Let the a be first term and d be common difference of an AP

Then sum of first n terms of an AP:

Given

S_{5 +} S₇ = 167

5[2a + 4d] + 7[2a + 6d] = 167(2)

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

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

23d = 115

d = 5

using this value in eqn 1

12a = 167 - 31(5)

12a = 167 - 155 = 12

a = 1

So, the AP is

a, a + d, a + 2d, a + 3d, … ,

1, 6, 11, 16

In an AP, if S₅ + S₇ = 167 and S₁₀ = 235, then find the AP, where S_n denotes the sum of its first n terms.