The geometric mean of a and b is \( \sqrt{ab}\) .
According to conditions given in the question,
\( \dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}\)
Square on both the sides.
\( \dfrac{(a^{n+1}+b^{n+1})^2}{(a^n+b^n)^2}=ab\)
\( a^{2n+2}+2a^{n+1}b^{n+1}+b^{2n+2}\)
= \( (ab)(a^{2n}+2a^nb^n+b^{2n})\)
\( a^{2n+2}+2a^{n+1}b^{n+1}+b^{2n+2}\)
= \( a^{2n+2}+2a^{n+1}b^{n+1}+ab^{2n+1}\)
= \( a^{2n+2}+b^{2n+2}=a^{2n+1}+ab^{2n+1}\)
\( a^{2n+2}-b^{2n+1}b=ab^{2n+1}-b^{2n+2}\)
= \( a^{2n+1}-=(a-b)=b^{2n+1}(a-b)\)
\(( \dfrac{a}{b})^{2n+1}=1=(\dfrac{a}{b})^0\)
\( 2n+1=0\)
\( n=\dfrac{-1}{2}\)
Answered by Abhisek | 1 year agoConstruct a quadratic in x such that A.M. of its roots is A and G.M. is G.
Find the geometric means of the following pairs of numbers:
(i) 2 and 8
(ii) a3b and ab3
(iii) –8 and –2
Insert 5 geometric means between \( \dfrac{32}{9}\) and \( \dfrac{81}{2}\).