Question

Given that the sum of the squares of the first seven natural numbers is 140, then their mean is ______.

Options

• 20

• 70

• 280

• 980

Answer 1

Given that the sum of the squares of the first seven natural numbers is 140, then their mean is 20.

Explanation:

Given:

• The sum of the squares of the first seven natural numbers is 140.

First, let's recall the formula for the sum of the squares of the first n natural numbers:

`S = sum_(i = 1)^ni^2`

= `(n(n + 1)(2n + 1))/6`

For the first seven natural numbers (n = 7):

`S = (7(7 + 1)(2 * 7 + 1))/6`

= `(7 * 8 * 15)/6`

Calculating the sum:

`S = (7 * 8 * 15)/6`

= `840/6`

= 140

The sum of the squares is indeed 140, which matches the given information.

To find the mean of these squares, we divide the sum by the number of natural numbers:

Mean = `"Sum of the square"/"Number of terms"`

Here, the number of terms is 7:

Mean = `140/7`

= 20

So, the mean of the squares of the first seven natural numbers is 20.