2.1 Change in Arithmetic and Geometric Sequences

Arithmetic Sequences: Constant Difference

An arithmetic sequence changes by a constant amount each step — the common difference (d). Consecutive terms show equal successive differences: $a_n - a_{n-1} = d$. This means arithmetic sequences grow (or shrink) linearly, and can be modeled by $a_n = a_0 + dn$, a linear function of $n$.

Geometric Sequences: Constant Ratio

A geometric sequence changes by a constant factor each step — the common ratio (r). Consecutive terms show equal successive quotients: $\frac{a_n}{a_{n-1}} = r$. This produces exponential growth or decay, modeled by $a_n = a_0 \cdot r^n$.

Recognizing Patterns from Data

To identify a sequence type, compute differences and ratios of consecutive terms. If differences are constant, it's arithmetic. If ratios are constant, it's geometric. Neither constant means the sequence is neither — this comparison is key for modeling real data sets with the right function type.

Key terms

Arithmetic sequence

A sequence with a constant common difference (d) between consecutive terms; models linear growth.

Geometric sequence

A sequence with a constant common ratio (r) between consecutive terms; models exponential growth/decay.

Common difference (d)

The constant value added/subtracted between consecutive terms in an arithmetic sequence: d = a_n - a_(n-1).

Common ratio (r)

The constant value multiplied between consecutive terms in a geometric sequence: r = a_n / a_(n-1).

How to test a sequence type

Check successive differences (constant → arithmetic) and successive ratios (constant → geometric).

Try an AP-style question

For the sequence 3, 7, 11, …, what is a₁₀?

1. 40
2. 43
3. 39
4. 35