Concept 5.5
Sum of the First n Terms
Source: NCERT Mathematics, Class 1012 minute lesson
Formal statement
The sum of the first n terms of an AP is Sₙ = n/2 [2a + (n − 1)d]. If the last term is l, this is also Sₙ = n/2 (a + l).
The intuition
Write the sum forwards, then backwards underneath. Every column adds to the same total: the first term plus the last.
So the sum is just how many pairs times what each pair adds to. The bracket is a + l in disguise, because l = a + (n − 1)d.
The attention
Look at the bracket: it is one pair, the first term plus the last.
Sₙ = n/2 [2a + (n − 1)d]
Look hereNot here
- Look here
- The bracket, one pair's total. 2a + (n − 1)d is a + l: the first term plus the last. Every pair, working inwards, adds to the same total.
- Not here
- n/2, the number of pairs. “Half of n” makes it look as if an odd n breaks the formula. It doesn't: n/2 counts pairs in the sum written twice, so it works for every n.
- What it teaches
- When a long sum appears, look for pairs that add to the same thing.