Arithmetic & Geometric Sequences
Master the two great families of sequences: find nth terms with explicit formulas, work recursions, and tell arithmetic from geometric at a glance.
1 What is a sequence?
A sequence is an ordered list of numbers: a1, a2, a3, … Each number is a term, and the subscript tells you its position. Two families run the show:
Arithmetic sequences add a constant common difference d each step. Geometric sequences multiply by a constant common ratio r each step. The exponent is n − 1, not n — the first term already uses exponent 0.
2 Explicit vs. recursive formulas
An explicit formula jumps straight to term n. A recursive formula crawls term by term from the one before it.
Explicit: d = 4, so a20 = 5 + (20 − 1)(4) = 5 + 76 = 81 — one step.
Recursive: a1 = 5, an = an−1 + 4 — you would add 4 nineteen times. Recursive is fine for describing the pattern; explicit is faster for finding a far-out term.
A recursive formula is useless without the starting term a1 — always state it.
3 Worked examples
Find the term, name the family, or recover the common difference or ratio.
Find a10 if a1 = 3 and d = 4.
a10 = 3 + (10 − 1)(4) = 3 + 36 = 39.
Find a5 if a1 = 2 and r = 3.
a5 = 2 · 35−1 = 2 · 81 = 162.
Is 5, 8, 11, 14, … arithmetic or geometric?
Differences: 3, 3, 3 — constant difference, so arithmetic with d = 3.
Is 3, 6, 12, 24, … arithmetic or geometric?
Ratios: 2, 2, 2 — constant ratio, so geometric with r = 2.
The arithmetic sequence begins 7, 2, −3, …. Find d.
d = 2 − 7 = −5. (Check: −3 − 2 = −5. Consistent.)
The geometric sequence begins 4, 12, 36, …. Find r.
r = 12 / 4 = 3. (Check: 36 / 12 = 3. Consistent.)
Given a1 = 2 and an = 3 · an−1, find a4.
a2 = 6, a3 = 18, a4 = 54.
4 Common mistakes
Two mix-ups to eliminate for good.
Wrong: crawling to a50 one addition at a time.
Right: explicit jumps straight there: a50 = a1 + 49d. Save recursion for describing patterns, not distant terms.
Wrong: calling 2, 6, 18, 54 arithmetic because “it keeps growing.”
Right: test it — constant difference → arithmetic; constant ratio → geometric. Here 6/2 = 18/6 = 54/18 = 3, so geometric.
5 Key vocabulary
Words to know
- Term (an) — the nth number in the sequence.
- Common difference (d) — the constant added each step in an arithmetic sequence.
- Common ratio (r) — the constant multiplier each step in a geometric sequence.
- Explicit formula — computes an directly from n.
- Recursive formula — computes an from an−1; needs a1 to start.
6 Quick check
Try these before moving on — click to reveal each answer.
7 Next steps
Now drill the skill with endless randomized problems.