If $x=\frac{1}{5}+\frac{1 \times 3}{5 \times 10}+\frac{1 \times 3 \times 5}{5 \times 10 \times 15}+\ldots$. then $3 x^2+6 x=$

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Aicharya · Accepted Answer

We have, $x=\frac{1}{5}+\frac{1 \cdot 3}{5 \cdot 10}+\frac{1 \cdot 3 \cdot 5}{5 \cdot 10 \cdot 15}+\ldots$ Comparing expression with, $(1+y)^n=\frac{1+n y}{1 !}+\frac{n(n-1)}{2 !} y^2+\ldots .$ we get, $n y=\frac{1}{5}, \frac{n(n-1)}{2 \cdot 1} \cdot y^2=\frac{1 \cdot 3}{5 \cdot 10}$ Solving we get, $y=-\frac{2}{5} \text { and } n=-\frac{1}{2}$ So, $\begin{array}{ll} x=(1+y)^n-1 \ \Rightarrow & x=\left(1-\frac{2}{5}\right)^{-\frac{1}{2}}-1=\sqrt{\frac{5}{3}}-1 \end{array}$ So, $\quad 3 x^2+6 x$ $\begin{aligned} & =3\left(\sqrt{\frac{5}{3}}-1\right)^2+6\left[\left(\sqrt{\frac{5}{3}}\right)-1\right] \ & =3\left(\frac{5}{3}+1-2 \sqrt{\frac{5}{3}}\right)+6 \sqrt{\frac{5}{3}}-6 \ & =5+3-6=2 \end{aligned}$

If \(x=\frac{1}{5}+\frac{1 \times 3}{5 \times 10}+\frac{1 \times 3 \times 5}{5 \times 10 \times 15}+\ldots\)…

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