Question

find the value of n so that [a^(n+1) + b^(n+1)]/[a^n + b^n] is the GM between a and b​

Answer 1

Answer:

The G. M between a and b isv(ab)

a^(n+1)+b^(n+1)/(a^n+b^n)=v(ab)then n=-1/2

Answer 2

$\purple{\underline{ \huge{\sf \fbox{Solution : \: }}}}$

We know that , the geometric mean of any two positive numbers a and b is given by √ab

Thus ,

$\large{\star \: } \mathtt{ \fbox{ \frac{ {a}^{(n + 1)} \: + \: {b}^{(n + 1)} }{ {a}^{n} \: + \: {b}^{n} } = \sqrt{ab} }}$

Squaring both sides , we get

$\sf \hookrightarrow \frac{ {a}^{(n + 1)} \: + \: {b}^{(n + 1)} }{ {a}^{(n)} \: + \: {b}^{(n)} } = \sqrt{ab} \\ \\ \sf \hookrightarrow {({\frac{ {a}^{(n + 1)} \: + \: {b}^{(n + 1)} }{ {a}^{(n)} \: + \: {b}^{(n)} }})}^{2} = ab \\ \\\sf \hookrightarrow {\frac{ {({a}^{(n + 1)} \: + \: {b}^{(n + 1)} )}^{2} }{ {({a}^{(n)} \: + \: {b}^{(n)} )}^{2} }} = ab \\ \\\sf \hookrightarrow \frac{{ {a}^{(n + 1)} }^{2} + { {(b)}^{(n + 1)} }^{2} + 2 {(a)}^{(n + 1)} {(b)}^{(n + 1)}}{{ {a}^{(n)} }^{2} + { {b}^{(n)} }^{2} + 2 {a}^{(n)} {b}^{(n)} } = ab \\ \\\sf \hookrightarrow \frac{ {a}^{(2n + 2)} + {b}^{(2n + 2)} + 2 {a}^{(n + 1)} {b}^{(n + 1)} }{ {a}^{(2n)} + {b}^{(2n)} + 2 {a}^{(n)} {b}^{(n)} } = ab \\ \\ \sf \hookrightarrow {a}^{(2n + 2)} + {b}^{(2n + 2)} + 2 {a}^{(n + 1)} {b}^{(n + 1)} = ab( {a}^{(2n)} + {b}^{(2n)} + 2 {a}^{(n)} {b}^{(n)} ) \\ \\\sf \hookrightarrow {a}^{(2n + 2)} + {b}^{(2n + 2)} + 2 {a}^{(n + 1)} {b}^{(n + 1)} = {a}^{(2n + 1)} b + a {b}^{(2n + 1)} + 2 {a}^{(n + 1)} {b}^{(n + 1)} \\ \\\sf \hookrightarrow {a}^{(2n + 2)} + {b}^{(2n + 2)} = {a}^{(2n + 1)} b + a {b}^{(2n + 1)} \\ \\ \sf \hookrightarrow {a}^{(2n + 2)} - {a}^{(2n + 1)} b = a {b}^{(2n + 1)} - {b}^{(2n + 2)} \\ \\\sf \hookrightarrow {a}^{(2n + 1)} (a - b) = {b}^{(2n + 1)} (a - b) \\ \\ \sf \hookrightarrow {a}^{(2n + 1)} = {b}^{(2n + 1)} \\ \\ \sf \hookrightarrow {(\frac{a}{b} )}^{(2n + 1)} = 1 \\ \\\sf \hookrightarrow {( \frac{a}{b})}^{(2n + 1)} = { (\frac{a}{b} )}^{(0)}$

By comparing the powers , we get

$\sf \hookrightarrow 2n + 1 = 0 \\ \\ \sf \hookrightarrow n = - \frac{1}{2}$

Hence , the required value of n is -1/2