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Question

Sum of infinite terms A,AR,AR2,AR3, is 15. If the sum of the squares of these terms is 150, then find the sum of AR2,AR4,AR6,.


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Solution

Step-1 : Applying the formula of the sum of an infinite geometric series

Formula to be used : We know that the sum of an infinite geometric series a,ar,ar2,ar3, with common ratio r is a1-r.

Here, the given series is A,AR,AR2,AR3,which is an infinite geometric series with common ratio R.. Then its sum will be A1-R.

Now, by the question, A1-R=15. 1

Step-2 : Finding the sum of the squares of the terms of the series A,AR,AR2,AR3,

After squaring each term, we get the series as : A2,A2R2,A2R4,A2R6, which is also an infinite geometric series with common ratio R2. So, its sum will be A21-R2. Now, by the question, A21-R2=150. 2

Step-3 : Finding the value of A

From Step-1 and Step-2, we get :

A21-R2=150A.A1-R1+R=1501-R2=1+R1-RA1-R.A1+R=150A1+R=15015A1-R=15A=101+R

Now, Substituting A=101+R in 1, we obtain :

A1-R=15101+R1-R=151+R1-R=323-3R=2+2R5R=1R=15

Hence,

A=101+R=101+15=10×65=12

Step-4 : Finding the sum of AR2,AR4,AR6,

The series AR2,AR4,AR6, is an infinite geometric series with common ratio R2. So, its sum will be

AR21-R2=12.1521-152=1224=12

Hence, the sum of AR2,AR4,AR6,=12.


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