Arithmetic & Geometric Sequences
“Add and it's arithmetic (a line); multiply and it's geometric (a curve).”
The formula
aₙ = a₁ + (n−1)d, aₙ = a₁·rⁿ⁻¹How to read it: an arithmetic term adds the common difference d to the first term (n−1) times; a geometric term multiplies the first term by the common ratio r (n−1) times
- a₁
- — the first term — where the sequence starts
- d
- — the common difference — the constant gap between neighbours (arithmetic)
- r
- — the common ratio — the constant factor between neighbours (geometric)
- n
- — the term number (which term it is)
The hook
Add the same amount each step, like climbing stairs → arithmetic. Multiply by the same factor, like interest compounding → geometric. One rule decides the whole sequence.
In plain words
An arithmetic sequence adds a fixed number d to the previous term each time; a geometric sequence multiplies the previous term by a fixed number r each time.
The intuition
Arithmetic walks with a constant stride — plotted, it's a straight line. Geometric leaps by the same factor each time — plotted, it's an exponential curve. Whether the rule is 'add' or 'multiply' is what separates them.
How it's built
To reach the n-th term you take exactly (n−1) steps from the first term. Arithmetic adds d per step → a₁+(n−1)d; geometric multiplies by r per step → a₁·rⁿ⁻¹. The key is (n−1), not n — the first term is zero steps.
Example
2, 5, 8, 11, … is arithmetic with d=3. The 4th term = 2 + (4−1)·3 = 2 + 9 = 11. And 3, 6, 12, 24, … is geometric with r=2. The 4th term = 3·2³ = 24.
Common mistake
A common slip is putting n straight into the exponent. The ratio factor for the 4th term is r³, not r⁴ — you multiply only three times from the first term. Always check the (n−1).
Where it's used
Savings interest, population growth, depreciation (geometric); fixed deposits, seat layouts (arithmetic) — anywhere a quantity changes by a regular rule and you want to predict ahead.
Where it came from
The tale of young Gauss adding 1 through 100 in a flash is exactly the sum of an arithmetic sequence — pairing the ends and adding.
Prerequisites
Quick check
In the arithmetic sequence 3, 7, 11, …, what is the 5th term?
- 15
- 17
- 19✓
- 23
Practice
Find the 10th term of the arithmetic sequence 2, 5, 8, ….
Answer: 29
- a₁=2, d=3
- a₁₀ = 2 + (10−1)·3 = 2 + 27 = 29
Takeaway: The n-th term adds d to the first term (n−1) times.
Find the 7th term of the geometric sequence 1, 2, 4, 8, ….
Answer: 64
- a₁=1, r=2
- a₇ = 1·2⁷⁻¹ = 2⁶ = 64
Takeaway: The n-th ratio factor is rⁿ⁻¹ (not r⁷).
An arithmetic sequence has first term 5 and common difference −2. Find the 6th term.
Answer: -5
- a₆ = 5 + (6−1)·(−2) = 5 − 10
- = −5
Takeaway: A negative common difference makes the sequence shrink.
Find the 4th term of a geometric sequence with first term 2 and common ratio 3.
Answer: undefined
- a₄ = 2·3⁴⁻¹ = 2·3³
- = 2·27 = 54
Takeaway: Multiply just three times from the first term.