Question

$\sqrt{1+2\left(0.4\right)+3{\left(0.4\right)}^2+4{\left(0.4\right)}^2+.....\infty }$ has the value equal to

• A. $\frac{5}{3}$
• B. $\frac{5}{4}$
• C. $1$
• D. None of these

Answer 1

The correct option is A $\frac{5}{3}$

Let $S=1+2\left(0.4\right)+3(0.4{)}^2+......$ ______ (1)

$\left(0.4\right)S=1\left(0.4\right)+2(0.4{)}^2+......$ _________ (2)

(1) - (2)

$\Rightarrow \left(0.6\right)S=1+\left(\right(0.4)+(0.4{)}^2+.....)$

$=1+\frac{\left(0.4\right)}{1-0.4}$

$=1+\frac{0.4}{0.6}$

$0.6S=\frac{5}{3}$

$S=(\frac{5}{3}{)}^2$

$\therefore$ Required Answer = $\sqrt{S}=\frac{5}{3}$