Question

The sum of all numbers of the form $n^3$ which lie between $100$ and $10,000$ is

• A. 43261
• B. 53261
• C. 63261
• D. none of these

Answer 1

Answer: (B)

The smallest and the largest numbers between $100$ and $10,000$ which can be written in the form $x^3$ are $5^3=125$ and $21^3=9261$
$\therefore$ the reqd. sum $=5^3+6^3+7^3+.......+21^3$
${\sum }_{n=1}^{21}-{\sum }_{n=1}^4{n}^3$
$={\left(\frac{n^2{\left(n+1\right)}^2}{4}\right)}_{n=21}-{\left(\frac{n^2{\left(n+1\right)}^2}{4}\right)}_{n=4}$
$\frac{441\times 484}{4}-\frac{16\times 25}{4}$
$=441\times 121-100$
$=53361-100$
$=53261$