Question

$1^3-2^3+3^3-4^3+...+9^3$ equals

• A. 425
• B. 400
• C. 405
• D. 395

Answer 1

The correct option is A 425

Let, $S=1^3-2^3+3^3-4^3+...+9^3$
$\Rightarrow S=(1^3+3^3+5^3+...+9^3)-(2^3+4^3+6^3+8^3)$
$\Rightarrow S=(1^3+3^3+5^3+...+9^3)-2^3(1^3+2^3+3^3+4^3)$
$\Rightarrow S={\sum }_{n=1}^5{(2n-1)}^3-2^3{\sum }_{n=1}^4{n}^3$
$\Rightarrow S=8{\sum }_{n=1}^5{n}^3-12{\sum }_{n=1}^5{n}^2+6{\sum }_{n=1}^{5}n-{\sum }_{n=1}^{5}1-2^3{\sum }_{n=1}^4{n}^3$

$\Rightarrow S=8\left[\frac{n^2(n+1{)}^2}{4}\right]-12\left[\frac{n(n+1)(2n+1)}{6}\right]+6\left[\frac{n(n+1)}{2}\right]-5-2^3\left[\frac{n^2(n+1{)}^2}{4}\right]$
$\Rightarrow S=8\left[\frac{5^2{(5+1)}^2}{4}\right]-12\left[\frac{5(5+1)(2.5+1)}{6}\right]+6\left[\frac{5(5+1)}{2}\right]-5-2^3\left[\frac{4^2{(4+1)}^2}{4}\right]$
$\Rightarrow S=1800-660+90-5-800=425$
Ans: A