The sum to 20 terms of the series 2 · 2²-3²+2 · 4²-5²+2 · 6²-… … is equal to __________.

Solution

Maths sir · 4 steps

Pair each even term with the odd term after it, and 20 terms become 10 neat pairs.

1. Spot the pairs. The terms come in pairs: 2 · 2² - 3², then 2 · 4² - 5², and so on. The k-th pair is 2(2k)² - (2k+1)², and 20 terms make exactly 10 pairs.
2. Simplify one pair. 2(2k)² - (2k+1)² = 8k² - (4k² + 4k + 1) = 4k² - 4k - 1.

   S = Σₖ₌₁¹⁰(4k² - 4k - 1) = 4Σₖ₌₁¹⁰ k² - 4Σₖ₌₁¹⁰ k - 10

3. Put in numbers. Σ k² = (10 × 11 × 21)/6 = 385 and Σ k = (10 × 11)/2 = 55. So S = 4 × 385 - 4 × 55 - 10 = 1540 - 220 - 10.
4. So. The sum to 20 terms is 1310.

The common slip: 10620. Running the pair formula from k = 1 to 20 instead of 10 gives 10620. Each pair uses two terms, so 20 terms means only 10 pairs.

The idea to keep. When signs or coefficients repeat in a cycle, group one full cycle into a single general term. Then sum that term with the standard formulas for Σ k and Σ k².

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