Sequences and the nth Term

Linear (arithmetic) sequences

A fixed amount (the common difference, d) is added each time.

The nth term formula

nth term = dn + (a − d)

where a = first term, d = common difference.

Sequence 5, 8, 11, 14… (a = 5, d = 3):

nth term = 3n + (5 − 3) = 3n + 2

Check: n = 1 → 3(1) + 2 = 5 ✓. 50th term = 3(50) + 2 = 152.

Is a number in the sequence?

Is 101 a term of 3n + 2? Solve 3n + 2 = 101 → n = 33. Yes — it's the 33rd term. (If n isn't a whole number, it isn't in the sequence.)

Quadratic sequences

The second difference is constant. nth term contains an n² term.

• 2, 5, 10, 17 → first differences 3,5,7; second difference 2 → contains n²: nth term = n² + 1.

Special sequences

Name	Sequence	nth term
Square	1, 4, 9, 16	n²
Cube	1, 8, 27, 64	n³
Triangular	1, 3, 6, 10	n(n+1)/2
Fibonacci	1,1,2,3,5,8	add previous two
Geometric	2,6,18,54	multiply by a ratio

Exam tip

For linear sequences: nth term = (difference)n + (first term − difference). A constant second difference always means a quadratic sequence.