Three numbers in AP have sum 15 and product 80. If they are a-d, a, a+d, then a = x.

Solution

Maths sir · 4 steps

Write the AP as a−d, a, a+d and the d's cancel when you add, so the sum gives a at once.

1. Add the terms. In the sum, -d and +d cancel, so only the three copies of a remain.

   (a-d) + a + (a+d) = 3a = 15

2. Solve for a. Dividing by 3 gives a = 5. The product is not needed to find a.
3. Check with product. The product gives 5(25 - d²) = 80, so d² = 9 and d = ± 3. The numbers are 2, 5, 8, whose product is 80.
4. So. The middle term is a = 5.

The common slip: 3. Reporting the common difference d = 3 from the product step gives 3, but the question asks for the middle term a.

The idea to keep. For three terms in AP with a given sum, take them as a-d, a, a+d. The sum gives a straight away, and then the product gives d.

Answer from the NTA final answer key. Solution by PrepWiser.