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 + … to ∞
We observe that the above progression possess a common ratio. So it is a geometric progression.
Common ratio = rp 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 ⇒ |rp|<1 if (p>1)
∴ we can use the formula for the sum of infinite GP.
⇒ Sp = ….equation 1
As, sp = 1 – rp + r2p - … to ∞
We observe that the above progression possess a common ratio. So it is a geometric progression.
Common ratio = -rp 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 ⇒ |rp|<1 if (p>1)
∴ we can use the formula for the sum of infinite GP.
⇒ sp = ….equation 2
As we have to prove - sp + Sp = 2 s2p
From equation 1 and 2, we get-
∴ Sp + sp =
⇒ Sp + sp = {using (a+b)(a-b)=a2-b2}
⇒ Sp + sp =
As Sp =
∴ following the same analogy, we have-
∴ Sp + sp =
Hence,
Sp + sp = 2S2p