Question

Assertion :For $r\ge 1$, and $x\ne 1$, let $t_r=1+2x+3x^2+\cdots +r{x}^{r-1}$.

Sum of $t_1+t_2+\cdots +t_n$ is $\frac{n(1+x^{n+1})}{(1-x{)}^2}-\frac{2x(1-x^n)}{(1-x{)}^3}$

Reason: For $r\ge 1$, and $x\ne 1,1+x+x^2+\cdots +x^{r-1}=\frac{1-x^r}{1-x}$ and $1+2x+3x^2+\cdots +r{x}^{r-1}=\frac{1-x^r}{(1-x{)}^2}-\frac{r{x}^r}{1-x}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

According to theory
$t_r=\frac{1-x^r}{(1-x{)}^2}-\frac{r{x}^r}{1-x}$
${\sum }_{r=1}^n{t}_r=\frac{1}{(1-x{)}^2}{\sum }_{r=1}^n(1-x^r)-\frac{1}{1-x}{\sum }_{r=1}^{n}r{x}^r$
$=\frac{1}{(1-x{)}^2}(n-\frac{x(1-x^n)}{1-x})-\frac{1}{1-x}(\frac{x(1-x^n)}{(1-x{)}^2}-\frac{n{x}^{n+1}}{1-x})$
$=\frac{n(1+x^{n+1})}{(1-x{)}^2}-\frac{2x(1-x^n)}{(1-x{)}^3}$