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प्रश्न
A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying the odd places. Find the common ratio of the G.P.
उत्तर
Let there be 2n terms in the given G.P. with the first term being a and the common ratio being r.
According to the question
Sum of all the terms = 5 (Sum of the terms occupying the odd places)
\[\Rightarrow a_1 + a_2 + . . . + a_{2n} = 5 \left( a_1 + a_3 + a_5 + . . . + a_{2n - 1} \right)\]
\[ \Rightarrow a + ar + . . . + a r^{2n - 1} = 5 \left( a + a r^2 + . . . + a r^{2n - 2} \right)\]
\[ \Rightarrow a\left( \frac{1 - r^{2n}}{1 - r} \right) = 5a\left\{ \frac{1 - \left( r^2 \right)^n}{1 - r^2} \right\} \]
\[ \Rightarrow 1 + r = 5 \]
\[ \therefore r = 4\]
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