Let s₁, s₂, s₃, …, s₁₀ respectively be the sum to 12 terms of 10 A.P. s whose first terms are 1,2,3, … .10 and the common differences are 1,3,5, … …, 19 respectively. Then Σi=1¹⁰ sᵢ is equal to :

Solution

Maths sir · 4 steps

Every sum is linear in a and d, so add all ten first terms and all ten differences together first.

1. Sum of one AP. For 12 terms, S = 12/2(2a + 11d) = 6(2a + 11d). This is linear in a and d, so adding ten such sums means adding the a's and the d's first.

   Σᵢ₌₁¹⁰ sᵢ = 6(2Σᵢ₌₁¹⁰ aᵢ + 11Σᵢ₌₁¹⁰ dᵢ)

2. Add a and d. First terms: 1 + 2 + … + 10 = 55. Common differences: 1 + 3 + … + 19 is the sum of the first 10 odd numbers, which is 10² = 100.
3. Put in numbers. 6(2 × 55 + 11 × 100) = 6 × 1210 = 7260.
4. So. Σ sᵢ = 7260, option (c).

The idea to keep. When many AP sums share the same number of terms, Sₙ is linear in a and d. Add all the first terms and all the differences once, then use the formula a single time.

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