Question

Find the mean of cubes of first 2P natural numbers

Answer 1

\frac{P(2P+1)^2}{2}

Answer 2

Solution Explanation:

1. The sum of cubes of the first n natural numbers is given by

$$\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2.$$

2. For $n = 2P$, the sum becomes

$$\left[\frac{2P(2P+1)}{2}\right]^2 = \left[P(2P+1)\right]^2.$$

3. The mean is the sum divided by $2P$:

$$\text{Mean} = \frac{\left[P(2P+1)\right]^2}{2P} = \frac{P(2P+1)^2}{2}.$$