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  4. If (1/2 × 4)+(1/4 × 6)+(1/6 × 8)+ ldots n terms =( kn / n +1), then k is equal to

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Q. If $\frac{1}{2 \times 4}+\frac{1}{4 \times 6}+\frac{1}{6 \times 8}+\ldots n$ terms $=\frac{ kn }{ n +1}$, then $k$ is equal to

Principle of Mathematical Induction

Solution:

$\frac{ kn }{ n +1}=\left[\frac{1}{2 \cdot 4}+\frac{1}{4 \cdot 6}+\frac{1}{6 \cdot 8}+\ldots n\right.$ terms $]$
$=\frac{1}{2}\left[\frac{4-2}{2 \cdot 4}+\frac{6-4}{4 \cdot 6}+\frac{8-6}{6 \cdot 8}+\ldots+\frac{2 n+2-2 n}{2 n(2 n+2)}\right]$
$=\frac{1}{2}\left[\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+\frac{1}{6}-\frac{1}{8}+\ldots+\frac{1}{2 n}-\frac{1}{2 n+2}\right]$
$=\frac{1}{2}\left[\frac{1}{2}-\frac{1}{2(n+1)}\right]=\frac{n}{4(n+1)} $
$\Rightarrow k=\frac{1}{4}$