If 1+(1–2².1)+(1–4².3)+(1-6².5)+......+(1-20².19)= α - 220β,
then an ordered pair ( α ,β ) is equal to:

Solution

Maths sir · 4 steps

Keep the lone 1 aside, write the k-th bracket in terms of k, and the rest is just Σ k³ and Σ k².

1. General term. After the first 1, the brackets follow the pattern 1 - (2k)²(2k-1). The last bracket has 2k = 20, so k runs from 1 to 10. That gives 10 brackets.
2. Count the ones. The standalone 1 and the ten 1's inside the brackets add to 11. The remaining part is Σₖ₌₁¹⁰ (2k)²(2k-1) = Σₖ₌₁¹⁰ (8k³ - 4k²).
3. Standard sums. Σₖ₌₁¹⁰ k³ = ((10 × 11)/2)² = 3025 and Σₖ₌₁¹⁰ k² = (10 × 11 × 21)/6 = 385.

   S = 11 - (8 × 3025 - 4 × 385) = 11 - (24200 - 1540) = 11 - 22660

4. So. 22660 = 220 × 103, so S = 11 - 220 × 103. This gives (α, β) = (11, 103), option (a).

The idea to keep. For a long sum of brackets, find the general term in k, expand it into powers of k, and use the standard formulas for Σ k, Σ k², Σ k³. Count the constant terms carefully.

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