An algebra problem by Vinay Jain

Algebra Level 4

In a GP the ratio of the sum of the first 11 terms to the sum of the last 11 terms is 1/8 and the ratio of the sum of all terms without the first nine to the sum of all the terms without the last nine is 2. Then the number of terms in GP is:

40 34 38 36

Vinay Jain by Vinay Jain , 22, India

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1 solution

Maggie Miller
Jul 20, 2015

Let a a be the first term, r r be the ratio, and k k be the number of terms. We are given

a 1 − r 11 1 − r a r k − 11 1 − r 11 1 − r = 1 8 \displaystyle \frac{a\frac{1-r^{11}}{1-r}}{ar^{k-11}\frac{1-r^{11}}{1-r}}=\frac{1}{8} and a r 9 1 − r k − 9 1 − r a 1 − r k − 9 1 − r = 2. \displaystyle\frac{ar^9\frac{1-r^{k-9}}{1-r}}{a\frac{1-r^{k-9}}{1-r}}=2.

That is, r 9 = 2 r^9=2 and r k − 11 = 8 r^{k-11}=8 . Then r k − 11 = ( r 9 ) 3 = r 27 r^{k-11}=(r^{9})^3=r^{27} , so k − 11 = 27 k-11=27 . That is, k = 38 k=\boxed{38} .

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