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Divide 32 into 4 parts, which form the four terms of an arithmetic progression (AP), such that the product of the first and the fourth terms is 80. What are the four terms of the AP?

Answer :

Final answer:

To divide 32 into 4 parts that form an arithmetic progression (AP), we can use the formula for the nth term of an AP. We can write two equations to solve for the values of the first term and the common difference. Solving these equations will give us the four terms of the AP.

Explanation:

To divide 32 into 4 parts that form an arithmetic progression (AP) with the product of the first and fourth terms being 80, we can use the formula for the nth term of an AP: nth term = first term + (n-1) * common difference. Let's assume the first term is a, and the common difference is d. The four terms of the AP can be represented as a, a+d, a+2d, and a+3d. We are given that (a)(a+3d) = 80. Substituting the values into the expression, we get a(a+3d) = 80.

Simplifying this equation, we have a² + 3ad = 80.

We also know that a + (a+3d) + (a+2d) + (a+3d) = 32. Simplifying this equation, we have 4a + 8d = 32.

We now have two equations with two variables, a and d. Solving these equations simultaneously, we can find the values of a and d, which will give us the four terms of the AP.

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