Five numbers are in A.P. whose sum is 25 and product is 2520. If one of these five numbers is -1/2 , then the greatest number amongst them is:

Solution

Maths sir · 4 steps

Take the five terms symmetric about the middle one, and the sum hands you that middle term at once.

1. Symmetric terms. Write the numbers as a-2d, a-d, a, a+d, a+2d. The d's cancel in the sum, so 5a = 25 and a = 5.
2. Use the product. Pair the terms: (a-d)(a+d) = 25-d² and (a-2d)(a+2d) = 25-4d². The product is 5(25-d²)(25-4d²) = 2520, so the two brackets multiply to 504.

   (25-d²)(25-4d²) = 504 ⇒ 4d⁴ - 125d² + 121 = 0

3. Solve for d. This is a quadratic in d²: d² = (125 ± 117)/8, so d² = 1 or d² = 121/4. With d = ± 1 the terms are 3, 4, 5, 6, 7, and -1/2 is not among them, so d = ±11/2.
4. So. The numbers are -6, -1/2, 5, 21/2, 16. Check: (-6)(-1/2)(5)(21/2)(16) = 2520. The greatest number is 16, option (d).

The idea to keep. For an odd number of terms in A.P., centre them as a, a ± d, a ± 2d. The sum gives a directly, and the product pairs into differences of squares.

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