Question

The product of three geometric means between 5 and 125 will be. (a) 3125 (b) 15625 (c) 125 (d) 625 ​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{Numbers are 5 and 125}$

$\underline{\textsf{To find:}}$

$\textsf{Product of 3 geometric means between 5 and 125}$

$\underline{\textsf{Solution:}}$

$\textsf{Let}\;\mathsf{G_1,G_2,G_3}\;$

$\textsf{be the 3 geometric means between 5 and 125}$

$\textsf{Let r be the common ration}$

$\textsf{Then,}$

$\mathsf{G_1=5\,r}$

$\mathsf{G_2=5\,r^2}$

$\mathsf{G_3=5\,r^3}$

$\mathsf{125=5\,r^4}$

$\implies\boxed{\mathsf{r^4=25}}$

$\implies\boxed{\mathsf{r^2=5}}$

$\textsf{Product of geometric means}$

$\mathsf{=G_1{\times}G_2{\times}G_3}$

$\mathsf{=5\,r{\times}5\,r^2{\times}5\,r^3}$

$\mathsf{=125\,r^6}$

$\mathsf{=125(r^2)^3}$

$\mathsf{=125(5)^3}$

$\mathsf{=125(125)}$

$\mathsf{=15625}$

$\underline{\textsf{Answer:}}$

$\textsf{Product of 3 geometric means between 5 and 125}$

$\textsf{is 15625}$

$\textsf{Option(b) is correct}$

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(c) x² +18x -16 = 0

(d) x² +18x +16 = 0

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

If in an Geometric Progression

First term = a

Common Ratio = r

Then n th term of the Geometric Progression is

$= \sf{a \times {r}^{n - 1} }$

TO DETERMINE

The product of three geometric means between 5 and 125

CALCULATION

Two given numbers are 5 & 125

Three geometric mean are to placed between 5 & 125

Let r be the Common Ratio

Then the Geometric Progression is

$\sf{5 \: , 5r \: , \: 5 {r}^{2} , \: 5 {r}^{3} ,125}$

So 125 is 5th term of the Geometric Progression

So

$\sf{ \: 5 \times {r}^{4} = 125\: }$

$\implies \: \sf{ \: {r}^{4} = 25 \: }$

$\implies \: \sf{ \: {r}^{2} = \pm \: 5 \: }$

Since common ratio ( r ) is a real number

So

$\: \sf{ \: {r}^{2} \ne \: - 5 \: }$

Hence

$\: \sf{ \: {r}^{2} = \: 5 \: }$

RESULT

The product of three geometric means between 5 and 125

$\sf{ \: = 5r \: \times 5 {r}^{2} \times 5 {r}^{3} \: }$

$\sf{ \: = {5}^{3} \times {r}^{6} \: }$

$\sf{ \: = {5}^{3} \times {( {r}^{2}) }^{3} \: }$

$\sf{ \: = {5}^{3} \times {5}^{3} \: }$

$\sf{ \: = {5}^{6} \: }$

= 15625

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