1+3+5²+7+9²+… upto 40 terms is equal to

Solution

Maths sir · 4 steps

Split the 40 terms into two lists of 20: the squared ones and the plain ones, then add each list.

1. Spot the pattern. The bases are the odd numbers 1, 3, 5, 7, …, 79. Terms in odd positions are squared (1 = 1², 5², 9², …). Terms in even positions stay plain (3, 7, 11, …).
2. Squared terms. There are 20 of them, with bases 1, 5, 9, …, 77, that is 4k-3 for k = 1 to 20. Expand (4k-3)² = 16k² - 24k + 9 and sum each part.

   Σₖ₌₁²⁰(4k-3)² = 16 × (20 × 21 × 41)/6 - 24 × (20 × 21)/2 + 9 × 20 = 45920 - 5040 + 180 = 41060

3. Plain terms. 3 + 7 + 11 + … + 79 is an AP with 20 terms. Its sum is 20/2(3 + 79) = 820.
4. So. The total is 41060 + 820 = 41880, which is option (b).

The idea to keep. When a series alternates between two rules, split it into two separate series by position. Write a general term for each, then use Σ k and Σ k².

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