Consider two G.Ps. 2, 2², 2³, ..... and 4, 4², 4³, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is (2)^(225/8), then Σₖ₌₁ⁿ k(n - k) is equal to :

Solution

Maths sir · 4 steps

Multiply all the terms as powers of 2, set the GM exponent equal to 225/8, and n comes out as 20.

1. Product of terms. The first G.P. multiplies to 2^(1+2+…+60) = 2¹⁸³⁰. Since 4 = 2², the second G.P. multiplies to 2^(2(1+2+…+n)) = 2ⁿ⁽ⁿ⁺¹⁾.

   (1830 + n(n+1))/(60+n) = 225/8

2. Solve for n. Cross-multiplying gives 14640 + 8n² + 8n = 13500 + 225n, so 8n² - 217n + 1140 = 0. The discriminant is 47089 - 36480 = 10609 = 103², so n = (217 ± 103)/16, which is 20 or 57/8. Only n = 20 is a whole number.
3. Evaluate the sum. Σ k(n-k) = n · n(n+1)/2 - n(n+1)(2n+1)/6. With n = 20 this is 20 × 210 - (20 × 21 × 41)/6 = 4200 - 2870.
4. So. The sum is 1330, which is option (c). You can check it with the compact form (n-1)n(n+1)/6 = (19 × 20 × 21)/6 = 1330.

The idea to keep. When every term is a power of the same base, the geometric mean is just the average of the exponents. Write everything in that base first, then work with exponents only.

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