The mean of 10 numbers 7 × 8, 10 × 10, 13 × 12, 16 × 14, ....... is ____________.

Solution

Maths sir · 3 steps

Write the r-th term as a product of two APs, expand it, and use the standard sums.

1. General term. The first factors 7, 10, 13, 16 form an AP with difference 3, so the r-th one is 3r+4. The second factors 8, 10, 12, 14 have difference 2, so the r-th one is 2r+6.

   Tᵣ = (3r+4)(2r+6) = 6r² + 26r + 24

2. Add ten terms. Use Σ r² = (10 × 11 × 21)/6 = 385 and Σ r = (10 × 11)/2 = 55. The total is 6 × 385 + 26 × 55 + 24 × 10 = 2310 + 1430 + 240 = 3980.
3. So. Mean = 3980/10 = 398.

The common slip: 3980. Stopping at the sum gives 3980. Divide by the 10 numbers to get the mean.

The idea to keep. When each term is a product of two APs, write the r-th term as a quadratic in r. Then Σ r², Σ r and Σ 1 do all the work.

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