Question

For what value of n,$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}$is the geometric mean of a and b?

• A. $-\frac{1}{3}$
• B. $-\frac{1}{2}$
• C. $-\frac{1}{4}$
• D. $-\frac{1}{8}$

Answer 1

The correct option is B $-\frac{1}{2}$

$\frac{a^{n+1}+b^{n+1}}{a^n+b^n}=\sqrt{ab}=a^{\frac{1}{2}}{b}^{\frac{1}{2}}$ by given condition

$\therefore a^{n+1}+b^{n+1}=a^n{a}^{\frac{1}{2}}{b}^{\frac{1}{2}}+b^n{b}^{\frac{1}{2}}{a}^{\frac{1}{2}}$

$=a^{\frac{n+1}{2}}\sqrt{b}+b^{\frac{n+1}{2}}\sqrt{a}$

$\therefore a^{\frac{n+1}{2}}(\sqrt{a}-\sqrt{b})=b^{\frac{n+1}{2}}(\sqrt{a}-\sqrt{b})$

$\therefore a^{\frac{n+1}{2}}=b^{\frac{n+1}{2}}$

Above is possible only when $n+\frac{1}{2}=0$

$\therefore n=-\frac{1}{2}$

Hence, option B.