Find the value of n so that $$\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$$  may be the geometric mean between a and b.

Asked by Abhisek | 1 year ago |  68

##### Solution :-

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 ago

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