If S_p

Given,

S_p = 1 + r^p + r^2p + … to ∞

We observe that the above progression possess a common ratio. So it is a geometric progression.

Common ratio = r^p and first term (a) = 1

Sum of infinite GP = ,where a is the first term and k is the common ratio.

Note: We can only use the above formula if |k|<1

As, |r|<1 ⇒ |r^p|<1 if (p>1)

∴ we can use the formula for the sum of infinite GP.

⇒ S_p = ….equation 1

As, s_p = 1 – r^p + r^2p - … to ∞

We observe that the above progression possess a common ratio. So it is a geometric progression.

Common ratio = -r^p and first term (a) = 1

Sum of infinite GP = ,where a is the first term and k is the common ratio.

Note: We can only use the above formula if |k|<1

As, |r|<1 ⇒ |r^p|<1 if (p>1)

∴ we can use the formula for the sum of infinite GP.

⇒ s_p = ….equation 2

As we have to prove - s_p + S_p = 2 s_2p

From equation 1 and 2, we get-

∴ S_p + s_p =

⇒ S_p + s_p = {using (a+b)(a-b)=a²-b²}

⇒ S_p + s_p =

As S_p =

∴ following the same analogy, we have-

∴ S_p + s_p =

Hence,

S_p + s_p = 2S_2p