The sum 5^3+6^3+…+15^3 is equal to:

The correct answer is Option (4) → 14,300

The sum of the cubes of the first n natural numbers is \(\left(\frac{n(n+1)}{2}\right)^2\). A sum that starts at \(5^3\) is the sum up to \(15^3\) minus the sum up to \(4^3\).

• Sum up to \(15^3\): \(1^3+2^3+\ldots+15^3=\left(\displaystyle\frac{15\times16}{2}\right)^2=120^2=14400\)

• Sum up to \(4^3\): \(1^3+2^3+3^3+4^3=\left(\displaystyle\frac{4\times5}{2}\right)^2=10^2=100\)

• Required sum: \(5^3+6^3+\ldots+15^3=14400-100=14300\)

Tip: The sum of the first n cubes is the square of the sum of the first n numbers: \(1^3+2^3+\ldots+n^3=(1+2+\ldots+n)^2\).