Question

The sum of the series 1 + $\frac{1.3}{6}$ + $\frac{\mathrm{1.3.5}}{6.8}$ + ............∞ is

• A. 1
• B. 0
• C.
• D. 4

Answer 1

The correct option is D

4

Let, S = 1 + $\frac{1.3}{6}$ + $\frac{\mathrm{1.3.5}}{6.8}$ + .......∞

⇒ $\frac{S}{4}$ = $\frac{1}{4}$ + $\frac{1.3}{4.6}$ + $\frac{\mathrm{1.3.5}}{\mathrm{4.6.8}}$ + ..........∞

⇒$\frac{1}{2}$ - $\frac{S}{8}$ = $\frac{1}{2}$ - $\frac{1}{2}$.$\frac{1}{4}$ - $\frac{1}{2}$.$\frac{1.3}{4.6}$ - $\frac{1}{2}$$\frac{\mathrm{1.3.5}}{\mathrm{4.6.8}}$ - ∞

⇒ $\frac{1}{2}$ - $\frac{S}{8}$ = 1 - $\frac{1}{2}$ + $\frac{\frac{1}{2}(\frac{1}{2}-1)}{1.2}$ - $\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-2)}{\mathrm{1.2.3}}$ + $\frac{\frac{1}{2}(\frac{1}{2}-1)(\frac{1}{2}-1)(\frac{1}{2}-3)}{\mathrm{1.2.3.4}}$.........∞

⇒ $\frac{1}{2}$ - $\frac{S}{8}$ = ${(1-1)}^{\frac{1}{2}}$ = 0

⇒ $\frac{S}{8}$ = $\frac{1}{2}$ ⇒ S = 4.