Answer :

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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