The Nth Term Formula: Why (n-1)?
Unpack the formula a_n = a + (n-1)d to find the 20th term of an AP.
Equation
y = 1 + (x-1)*1
Graph
Table
| x | y |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | 10 |
| 11 | 11 |
| 12 | 12 |
| 13 | 13 |
| 14 | 14 |
| 15 | 15 |
| 16 | 16 |
| 17 | 17 |
| 18 | 18 |
| 19 | 19 |
| 20 | 20 |
What this lesson covers
What you do
You shape the function y = a + (x-1)*d and watch the graph answer.
Challenges to clear
- Set a = 5 and d = 3: the 20th term (x = 20) is 62 — that is 5 plus NINETEEN steps, not twenty.
- And the 1st term is just a itself: x = 1 shows 5 — zero steps taken.
- A new sequence starting at 7 — slide a there.
- Now the step: the 15th term must be 63. That is 7 plus FOURTEEN steps, not fifteen.
Check yourself
Why does the formula use (n-1) instead of just n?
If a = 10 and d = 4, which expression correctly finds the 5th term?
- Because the first term is already at position 1, so you only need n-1 steps to reach the nth term. — correct
- Because the common difference d is always subtracted from n in an increasing AP.
- Because the formula was derived using zero-based indexing from computer science.
- Because n represents the last term, so we subtract 1 to find the previous term.
- 10 + (5 - 1)*4 — correct
- 10 + 5*4
- 10 + (5 - 1) + 4
- 10 * (5 - 1)*4
Think about it
- a_n = a + (n−1)d: with a = 5 and d = 3, the 20th term takes 19 steps of 3. What is it?