Library / Foundations: Maths / FMA-04
Foundations: Maths
Arithmetic sequence: nth term
Term at position n when each term rises by a constant step.
- Ref
- FMA-04
- Rev
- v1
- Field
- Foundations: Maths
- Result
- un
a is the first term, d the common difference and n the position, starting at 1.
Start at the first term and add the common difference one fewer times than the position number.
Calculator
- Previous term35
- Sum of first n terms210
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The nth term is a straight-line formula in n, with gradient d.
- The position n−1 counts the steps, not n, because the first term needs no step.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| n Position | 2 to 50 | 7 | 199 | 192 | +10% | +10.3 % |
| d Common difference | 1 to 10 | 12 | 93 | 81 | +10% | +9.23 % |
| a First term | 1 to 20 | 37 | 56 | 19 | +10% | +0.769 % |
Sweep of Position
| Position | nth term |
|---|---|
| 2 | 7 |
| 8 | 31 |
| 14 | 55 |
| 20 | 79 |
| 26 | 103 |
| 32 | 127 |
| 38 | 151 |
| 44 | 175 |
| 50 | 199 |
Sweep one variable
Variables and how each one contributes
nth term
What the equation gives you. Everything else feeds it.
First term
The term at position 1. Shifts every term equally.
Common difference
Step between consecutive terms. Multiplies by (n − 1), so its effect grows with position.
Position
Which term you want. Linear: each extra position adds d.
Worked example with the values above
- Steps taken: n − 1 = 9.
- Added amount: 9 × 4 = 36.
- un = 3 + 36 = 39.
Assumptions and limits
- The difference between terms is constant.
- n is a whole number from 1.
Common mistakes
- Multiplying d by n instead of n − 1.
- Starting the count at 0.
Statistics and uncertainty
If d is estimated from data, its error is multiplied by (n − 1), so extrapolating to a far term magnifies the uncertainty.
Where it is used
- Spacing and scheduling patterns.
- Linear trends in data tables.
Maths you need
- Patterns: recognise a constant step
- Algebra: substitute into a formula