Question

The sum $\sum _{k=1}^{n}1+2+\cdots +k$ is a polynomial in $n.$ The degree of the polynomial is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$\sum _{k=1}^{n}1+2+\cdots +k$
$=\sum _{k=1}^n\frac{k(k+1)}{2}$
$=\frac{1}{2}\left(\sum _{k=1}^n{k}^2+\sum _{k=1}^{n}k\right)$

$=\frac{1}{2}(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2})$
which is a polynomial of degree $3.$