Question

If $\frac{a^{n+3}+b^{n+1}}{c^n+b^n}$ then

Answer 1

(a) By the given condition
$\frac{a^{n+1}+b^{n+2}}{a^n+b^n}=\frac{a+b}{2}$
or $2a^{n+1}+2b^{n+1}=a^{n+1}+b^{n+1}+a{b}^a+b{a}^n$
or $a^{n+1}-a^{n}b=b^{n}a-b^{n+1}$
or $a^a(a-b)=b^a(a-b)$
$\therefore a^n=b^n$. Above is possible only when $n=0$
$\because a^0=b^0=1$
(b) $\frac{a^{n+1}+b6n+1}{a^n+b^n}=\sqrt{ab}=a^{1/2}{b}^{1/2}$
$\therefore a^{n+1}+b^{n+1}=a^n{a}^{1/2}{b}^{1/2}+b^n{b}^{1/2}{a}^{1/2}$
$a^{n+1/2}\sqrt{b}+b^{n+1/2}\sqrt{a}$
$\therefore a^{n+1/2}(\sqrt{a}-\sqrt{b})=b^{n+1/2}(\sqrt{a}-\sqrt{b})$
$\therefore a^{n+1/2}=b^{n+1/2}$
Above is possible only when $n+\frac{1}{2}=0$
$\therefore =-\frac{1}{2}$
(c) $\frac{a^{n+1}+b^{n+2}}{a^n+b^n}=$ H.M. of a and b $=\frac{2ab}{a+b}$
$\therefore a.a^{n+1}+a{b}^{n+1}+b{a}^{n+1}+b.b^{n+1}$
$=2a^{n+1}b+2b^{n+1}a$
$\therefore a.a^{n+1}+b.b^{n+1}=a^{n+1}b+b^{n+1}a$
or $a^{n+1}(a-b)=b^{n+1}(a-b)$
or $a^{n+1}=b^{n+1}$
or ${\left(\frac{a}{b}\right)}^{n+1}=1\therefore n+1=0$ or $n=-1$