The Total solution for NCERT class 6-12
If Sp denotes the sum of the series 1 + rp +r2p + … to ∞ and sp the sum of the series 1 – rp +r2p – … to ∞, prove that sp + Sp =2 S2p.
Given:
Sp = 1+ rp + r2p + … ∞
By using the formula,
S∞ =a/(1 – r)
Where, a = 1, r = rp
So,
Sp = 1/ (1 – rp)
Similarly, sp =1 – rp + r2p – … ∞
Where, a = 1, r = -rp
Sp = 1/ (1 – (-rp))
= 1 / (1 + rp)
Now, Sp +sp = [1 / (1 – rp)] + [1 / (1 + rp)]
2S2p =[(1 – rp) + (1 + rp)] / (1 – r2p)
= 2 /(1 – r2p)
∴ 2S2p =Sp + Sp