Question

If $1+2x+{3x}^2+{4x}^3+\cdot \cdot \cdot \cdot \cdot \cdot \infty \ge 4$, then

• A. least value of $x$ is $\frac{1}{2}$
• B. greatest value of $x$ is $\frac{4}{3}$
• C. least value of $x$ is $\frac{2}{3}$
• D. greatest value of $x$ is $\frac{1}{2}$
  

Answer 1

Answer: (A)

least value of $x$ is $\frac{1}{2}$

Let sum be the $S$,
$S=1+2x+3x^2+..$
By multiplying above equation by $x$,
$Sx=x+2x^2+3x^3+..$
By subtracting above equations we get,
$S(1-x)=1+x+x^2...$
Using the formula,
$a+ar+a{r}^2...=\frac{a}{1-r}$
We get
$S(1-x)=\frac{1}{1-x}$
$S=\frac{1}{{(1-x)}^2}$
Given $S\ge 4$
$\frac{1}{{(1-x)}^2}\ge 4$
So we get,
$x\ge \frac{1}{2}$
Therefore, least value of $x$ is $\frac{1}{2}$
Hence, option 'A' is correct.