The four numbers in the arithmetic progression (AP) can be represented as a-d, a, a+d, a+2d, where 'a' is the middle term and 'd' is the common difference.

Given that one of these numbers is equal to the sum of squares of the other three numbers, we can write the following equation:

(a-d)² + a² + (a+d)² = (a+2d)²

Expanding and simplifying this equation, we get:

a² - 2ad + d² + a² + a² + 2ad + d² = a² + 4ad + 4d²

This simplifies to:

3a² + 2d² = a² + 4ad + 4d²

Subtracting a² + 2d² from both sides, we get:

2a² - 4ad = 2d²

Dividing by 2, we get:

a² - 2ad = d²

This implies that a = 2d.

Substituting a = 2d into the AP, we get the four numbers as -d, 2d, 5d, 8d.

The sum of these four numbers is -d + 2d + 5d + 8d = 14d.

So, the sum of all the four numbers is 14 times the common difference.

Question

The four numbers in the arithmetic progression (AP) can be represented as a-d, a, a+d, a+2d, where 'a' is the middle term and 'd' is the common difference.

Given that one of these numbers is equal to the sum of squares of the other three numbers, we can write the following equation:

(a-d)² + a² + (a+d)² = (a+2d)²

Expanding and simplifying this equation, we get:

a² - 2ad + d² + a² + a² + 2ad + d² = a² + 4ad + 4d²

This simplifies to:

3a² + 2d² = a² + 4ad + 4d²

Subtracting a² + 2d² from both sides, we get:

2a² - 4ad = 2d²

Dividing by 2, we get:

a² - 2ad = d²

This implies that a = 2d.

Substituting a = 2d into the AP, we get the four numbers as -d, 2d, 5d, 8d.

The sum of these four numbers is -d + 2d + 5d + 8d = 14d.

So, the sum of all the four numbers is 14 times the common difference.

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