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Question -

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.



Answer -

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 – … ∞

By using the formula,

S =a/(1 – r)

Where, a = 1, r = -rp

So,

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

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