Concept 5.6
From Sums Back to Terms
Source: NCERT Mathematics, Class 108 minute lesson
Formal statement
If Sₙ is the sum of the first n terms of an AP, then a₁ = S₁ and, for n ≥ 2, aₙ = Sₙ − Sₙ₋₁.
The intuition
A running total grows by one term at each step. So the amount it grows by is the newest term.
If Sₙ = n² + 2n, then S₃ = 15 and S₂ = 8, so the third term is 15 − 8 = 7. Dividing S₃ by 3 gives 5, the average of the first three terms: a different thing.
The attention
Look at how much the total grew: that growth is the newest term.
aₙ = Sₙ − Sₙ₋₁ not Sₙ ÷ n
Look hereNot here
- Look here
- Sₙ − Sₙ₋₁, how much the total grew. A running total grows by exactly one term each time. The size of that growth is the term that was just added.
- Not here
- Sₙ ÷ n, the average term. Sₙ ÷ n is the average of the first n terms. It equals the nth term only when every term is the same.
- What it teaches
- A total and its change are different things. The change tells you about the newest piece.