If S denotes the sum of an infinite G.P. and S1 denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively and

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 =


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