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: u_n

u_{n} = a + (n - 1) d

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

Solve for

u_nnth termComputed

aFirst term

120

dCommon difference

110

nPosition

250

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.

nth term () current value 39

nPositionKey driver

7199

dCommon difference

1293

aFirst term

3756

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

Vary

nth term ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

u_n

nth term

Result ·

What the equation gives you. Everything else feeds it.

First term

· typical 1 to 20

The term at position 1. Shifts every term equally.

Common difference

· typical 1 to 10

Step between consecutive terms. Multiplies by (n − 1), so its effect grows with position.

Position

· typical 2 to 50

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.
u_n = 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

Related formulas

Practise this formula