If $\frac{a^{n} + b^{n}}{a^{n - 1} + b^{n - 1}}$ is the A.M. between a, b then find the value of n.
OR
If there are $(2 n + 1)$ terms in A.P., then prove that the ratio of the sum of the odd terms and the even terms is $(n + 1) : n$ .

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Solution

Let $a, a + d, a + 2 d . . . a + (2 x + 1) \to R e_{2}$ series
The odd terms a, a+2d, a+4d
first term = a
common diff = 2d
No. of term = x+1
Sum of odd term $= \frac{(n + 1)}{2} (2 a + (n + 1 - 1) 2 d)$
$= (n + 1) (a + n d)$
The serises of even term..
a+d, a+3d, a+5d...
first term =a+d
No. of terms =x
common diff = 2d
Sum of even terms = $\frac{n}{2} [2 (a + d) + (n - 1) 2 d]$
$= \frac{h}{2} [2 a + 2 d + 2 n d - 2 d]$
$= h [a + n d]$
$\frac{5 o d d}{5 e v e n} = \frac{(n + 1) (a + n d)}{n (a + n d)} = (n + 1) : h$

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