Question

Sum of first and 25th terms of an arithmetic sequence is 70.
(iii) Find the sum of first 25 terms of this sequence.

Hint:

When you know the first and the last term of the sum you want to compute, always use the formula Sₙ = (n)/(2)(first term + last term). It's the most efficient method and avoids the need to find a and d separately.

Answer 1

Using the same AP, we need to find the sum of the first 25 terms (S₂₅).

There are two common formulas for the sum of an AP:
1. Sₙ = (n)/(2)[2a + (n-1)d]
2. Sₙ = (n)/(2)[a₁ + aₙ]
The second formula is more direct here since we are given information about a₁ and a₂₅.

We need to find S₂₅.
Using the formula Sₙ = (n)/(2)[a₁ + aₙ] with n=25:
S₂₅ = (25)/(2)[a₁ + a₂₅] We are given that a₁ + a₂₅ = 70.
Substituting this value into the sum formula:
S₂₅ = (25)/(2)(70) S₂₅ = 25 × 35 To calculate 25 × 35:
25 × 35 = 25 × (30 + 5) = (25 × 30) + (25 × 5) = 750 + 125 = 875 So, the sum of the first 25 terms is 875.

Alternative method using a₁₃:
We know the middle term is a₁₃ = 35. For an odd number of terms, the sum is the number of terms multiplied by the middle term.
S₂₅ = 25 × a₁₃ S₂₅ = 25 × 35 = 875 The sum of the first 25 terms is 875.