Solution:

Given:

S_p = 1 + r^p + r^2p + … ∞

By using the formula,

S_∞ = a/(1 – r)

Where, a = 1, r = r^p

So,

S_p = 1 / (1 – r^p)

Similarly, s_p = 1 – r^p + r^2p – … ∞

By using the formula,

S_∞ = a/(1 – r)

Where, a = 1, r = -r^p

So,

S_p = 1 / (1 – (-r^p))

= 1 / (1 + r^p)

Now, S_p + s_p = [1 / (1 – r^p)] + [1 / (1 + r^p)]

2S_2p = [(1 – r^p) + (1 + r^p)] / (1 – r^2p)

= 2 /(1 – r^2p)

∴ 2S_2p = S_p + S_p