Question

The sum of the series $1^3-2^3+3^3-.........+9^3=$

• A. $300$
• B. $123$
• C. $425$
• D. $0$

Answer 1

The correct option is C $425$

$1^3-2^3+3^3-4^3+5^3-6^3+7^3-8^3+9^3$

$=1^3+3^3+5^3+7^3+9^3-(2^3+4^3+8^3+6^3)$

$=1^3+3^3+5^3+7^3+9^3-2^3(1^3+2^3+4^3+3^3)$

Now, for

$1^3+3^3+5^3+7^3....+n^3=n^2({2n}^2-1)$

and, $1^3+2^3+3^3....+n^3=\frac{n^2{(n+1)}^2}{4}$

Using these expressions we can write

$=1^3+3^3+5^3+7^3+9^3-2^3(1^3+2^3+3^3+4^3)$

$=5^2(2{\left(5\right)}^2-1)-2.4^2{(4+1)}^2$

$=25\times 49-32\times 25$

$=25\times 17$

$=425$.