If S denotes the

Let a be the first term, and r be the common ratio.

According to the question-

a + ar + ar² + …∞ = S

⇒ S = a(1+r+r²+…∞)

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

Common ratio = r 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

∴ S = …equation 1

Also, as per the question

S₁ = a² + a²r² + a²r⁴ + …∞

⇒ S₁ = a² (1+r²+r⁴+…∞)

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

Common ratio = r² 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

∴ S₁ =

⇒ S₁ =

From equation 1,we have-

⇒ S₁ = ….equation 2

Dividing equation 1 by 2, we get-

⇒

⇒ (1-r)S² = (1+r)S₁

⇒ S² – S₁ = r(S² + S₁)

∴ r =

Put the value of r in equation 1 to get a.

a =