Thank you for reporting, we will resolve it shortly
Q.
The sum of all numbers of the form $n^3$ which lie between $100$ and $ 10,000$ is
Sequences and Series
Solution:
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\times484}{4} - \frac{16\times25}{4} $
$= 441 \times121 -100$
$ = 53361 -100 $
$= 53261$