Question

If
8 GM's are inserted between 3 and 4
then the product of 8 GM's​

Answer 1

SOLUTION

GIVEN

8 GM's are inserted between 3 and 4

TO DETERMINE

The product of 8 GM

EVALUATION

We have insert 8 GM( Geometric Mean) between 3 & 4

So we get a Geometric Progression with

First term = a = 4

Last term = l = 10 th term = 4

Let Common Ratio = r

The the progression is

$\sf{ 3, 3r , 3{r}^{2} , ...., {3r}^{8}, 4 \: }$

So by the condition

$\sf{ 3 \times {r}^{10 - 1} = 4 \: }$

$\implies \displaystyle\sf{ {r}^{9} = \frac{4}{3} \: }$

So the product of the Geometric means

$\sf{ = 3r\times 3{r}^{2}\times .... \times {3r}^{8} \: }$

$\sf{ = {3}^{8} \times {r}^{1 + 2 + ... + 8} \: }$

$= \sf{ {3}^{8} \times {r}^{36} \: }$

$\displaystyle \sf{ = {3}^{8} \times {\bigg( \frac{4}{3} \bigg)}^{4} \: }$

$= \sf{ {3}^{4} \times {4}^{4} }$

$= \sf{ {(3 \times 4)}^{4}}$

$= \sf{ {12}^{4}}$

FINAL ANSWER

If 8 GM's are inserted between 3 and 4

then the product of 8 GM $= \sf{ {12}^{4}}$

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