EQN-03-06

Sequences & Patterns

Spot arithmetic and geometric rules, then jump straight to the nth term without listing everything.

~5 min read•Difficulty 2/5

Arithmetic sequences (constant difference)

Each term adds a fixed amount d.

nth term = a_{1} + (n - 1) \cdot d

3, 7, 11, 15, \dots a_{1} = 3, d = 4 \to 10 th term = 3 + 9 \cdot 4 = 39

Geometric sequences (constant ratio)

Each term multiplies by a fixed ratio r.

nth term = a_{1} \cdot r^{n - 1}

2, 6, 18, 54, \dots a_{1} = 2, r = 3 \to 5 th term = 2 \cdot 3^{4} = 162

How to tell them apart

Check consecutive terms: a constant difference $\to$ arithmetic; a constant ratio $\to$ geometric. If neither, look for a figural or repeating pattern.

Don't list to the nth term

For large n, listing every term wastes time and invites slips. Use the formula. For repeating cycles (e.g. a pattern of period 4), use the remainder of $n \div ($ cycle length) to find the position.

Worked examples

Example 1 (difficulty 2/5)

In the sequence 5, 9, 13, 17, …, what is the 20th term?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A pattern repeats every 4 symbols: ◆ ● ▲ ★ ◆ ● ▲ ★ … What is the 27th symbol?

◎ Predict the answer first:

Tips

Test for a constant difference first (arithmetic), then a constant ratio (geometric).
Remember the nth term uses $(n - 1)$ steps, not n — an off-by-one is the usual error.
For cyclic patterns, divide n by the period and use the remainder.

Common mistakes

Using n instead of

(n - 1)

steps in the nth-term formula

Why: It feels like the term number should multiply d directly.

Fix: The first term takes 0 steps, so the nth term takes

(n - 1)

of them.

Treating a geometric sequence as arithmetic

Why: Both "grow".

Fix: Check whether terms differ by a constant amount (add) or a constant factor (multiply).

← Back to Algebra Practice this lesson →