Question

The ratio of sum of first 3 terms of a geometric progression to the sum of first 6 terms is 64:91. the common ratio of gp is

Answer 1

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Answer 2

Answer:

The common ratio of the gp is 3/4

Step-by-step explanation:

The sum of first n terms of a geometric series is given by

$S_n=\frac{a(r^n-1)}{r-1}$

Here a is the first term and r is the common ratio.

Now, we have been given that ratio of sum of first 3 terms of a geometric progression to the sum of first 6 terms is 64:91.

Hence, we have

$\frac{\frac{a(r^3-1)}{r-1}}{\frac{a(r^6-1)}{r-1}}=\frac{64}{91}$

Simplifying, we get

$\frac{r^3-1}{r^6-1}=\frac{64}{91}\\\\\frac{r^2+r+1}{\left(r+1\right)\left(r^4+r^2+1\right)}=\frac{64}{91}\\\\\left(r^2+r+1\right)\cdot \:91=\left(r+1\right)\left(r^4+r^2+1\right)\cdot \:64\\\\r=\frac{3}{4}$

Therefore, common ratio is 3/4