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A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then find its common ratio.

Grade:Upto college level

1 Answers

Latika Leekha
askIITians Faculty 165 Points
6 years ago
Let the G.P be A1, A2, A3, A4, …... , A2n.
The number of terms in the G.P is 2n. (Since, it is given that the G.P. has even number of terms).
Now, the sum of all the terms is 5 times the sum of terms occupying odd places, hence we have
(A1 + A2 + A3 + A4 + …... + A2n) = 5 (A1 + A3 + A5 + A7 + …... + A2n-1)
This yields (A1 + A2 + A3 + A4 + …... + A2n) - 5 (A1 + A3 + A5 + A7 + …... + A2n-1) = 0
(A2 + A4 + A6 + …... + A2n) = 4 (A1 + A3 + A5 + A7 + …... + A2n-1)
Now, if we assume the G.P. to be x, xd, xd2, …... , where ‘x’ is the first term and ‘d’ is the common ratio.
Then xd ( dn – 1) / (d-1) = 4x ( dn – 1) / (d-1)
This gives 4x = xd.
Hence, d = 4.
The common ratio of the given G.P. is 4.
Thanks & Regards
Latika Leekha
askIITians Faculty

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