Question

Let $(1+x{)}^n=1+nx+\frac{n(n-1)x^2}{2}+\cdots \cdots |x|$ < 1, n $ϵ$ $Z$ Then the sum of the series $3+\frac{8}{3}+\frac{80}{3^3}+\frac{240}{3^4}+.....$ is

• A. 9
• B. 27
• C. 12
• D. 101

Answer 1

The correct option is B 27

$(1+x{)}^n=3+\frac{8}{3}+\frac{80}{3^3}+\frac{240}{3^4}+\cdots$
$=1+nx+\frac{n(n-1)x^2}{2}+\cdots ie1+3\left(\frac{2}{3}\right)+6{\left(\frac{2}{3}\right)}^2+10{\left(\frac{2}{3}\right)}^3+15{\left(\frac{2}{3}\right)}^4+\cdots$