Concept 5.3
The nth Term
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
The nth term of an AP with first term a and common difference d is aₙ = a + (n − 1)d. It is also called the general term of the AP.
The intuition
To reach the nth term you start at a and jump. How many jumps? Not n. The first term needs zero jumps and the second needs one, so the nth needs n − 1.
It is the fence-post idea: five posts have four gaps between them. The (n − 1) is counting gaps, not posts.
The attention
Look at (n − 1): it counts jumps, not terms.
aₙ = a + (n − 1)d
Look hereNot here
- Look here
- The jump count, n − 1. It counts the jumps from the first term. The first term is zero jumps away, so the nth term is n − 1 jumps away.
- Not here
- The position, n. The position n is one more than the number of jumps. Using n gives a + nd, one step too far.
- What it teaches
- Count the moves, not the stops. Off-by-one mistakes come from counting the wrong thing.