Question

The first term of an arithmetic series of consecutive integers is $k^2+1$. The sum of 2k + 1 terms of this series may be expressed as:

• A. $k^3+(k+1{)}^3$
• B. $(k-1{)}^3+k^3$
• C. $(k+1{)}^3$
• D. $(k+1{)}^2$

Answer 1

Answer: (A)

$k^3+(k+1{)}^3$

First term of the series = $k^2+1$

Series : $k^2+1$, $k^2+2$, ................

$k^2(2k+1)+{\sum }_{i=1}^{2k+1}i=k^2(2k+1)+\frac{(2k+1)(2k+2)}{2}=(2k+1)(k^2+k+1)=2k^3+2k^2+2k+k^2+k+1=2k^3+3k^2+3k+1=k^3+k^3+3k^2+3k+1=k^3+{(k+1)}^3$