Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • NCERT
  • If A is the arithmetic mean and G1, G2 be two geometric means between any two numbers, then prove that 2Aequalsg1^2

If A is the arithmetic mean and G_1, G_2 be two geometric means between any two numbers, then prove that

2 \mathrm{A}=\frac{\mathrm{G}_{1}^{2}}{\mathrm{G}_{2}}+\frac{\mathrm{G}_{2}^{2}}{\mathrm{G}_{1}}

Answers (1)

It is given that A is the arithmetic mean and G_1, G_2  be two geometric means between any two numbers.


   \\ A=\frac{a+b}{2} \\\\ G=\sqrt {ab} \\\\ G_{1}=\sqrt {aG_{2}} \\\\ G_{2}=\sqrt {G_{1}b} \\\\ G_{1}^{2}=aG_{2} \\\\


 \\ G_{1}^{2}=a\sqrt {G_{1}b} \\\\ G_{1}^{4}=a^{2}G_{1}b \\\\ G_{1}^{3}=a^{2}b \\\\ G_{1}=a^{\frac{2}{3}}b^{\frac{1}{3}} \\\\
\\ G_{2}= \left( a^{\frac{2}{3}}b^{\frac{1}{3}} \times b \right) ^{\frac{1}{2}}=a^{\frac{1}{3}}b^{\frac{2}{3}}~ \\\\ \frac{G_{1}^{2}}{G_{2}}= \left( \frac{ \left( a^{\frac{2}{3}}b^{\frac{1}{3}} \right) ^{2}}{a^{\frac{1}{3}}b^{\frac{2}{3}}} \right) =a \\\\ \frac{G_{2}^{2}}{G_{1}}= \left( \frac{ \left( b^{\frac{2}{3}}a^{\frac{1}{3}} \right) ^{2}}{b^{\frac{1}{3}}a^{\frac{2}{3}}} \right) =b \\\\ \frac{G_{1}^{2}}{G_{2}}+\frac{G_{2}^{2}}{G_{1}}=a+b=2A \\\\

 

Posted by

infoexpert21

View full answer

Similar Questions

Latest Question