The three sequence types
A sequence is just an ordered list of numbers that follows a rule. The three types here cover the most common patterns you meet in algebra and finance.
An arithmetic sequence grows by adding the same amount each step, so the gap between neighbours is constant. A geometric sequence grows by multiplying by a fixed ratio, which makes it climb (or shrink) far faster. A Fibonacci-style sequence builds each term by adding the two before it.
- Arithmetic: add the common difference d each step.
- Geometric: multiply by the common ratio r each step.
- Fibonacci: each term is the sum of the previous two.
Reading the result
The nth term is the last value in the list — the value at the position you asked for. The sum adds every term from the first up to the nth, which is useful for series problems.
The chart traces the terms in order, making the shape of the growth obvious: arithmetic sequences plot as a straight line, while geometric and Fibonacci sequences curve upward steeply.
Practical tips
Choosing the right ratio or difference is what shapes the sequence. Small changes to the common value can have a dramatic effect once you take many terms.
- A negative common difference makes an arithmetic sequence count down.
- A ratio between 0 and 1 makes a geometric sequence shrink toward zero.
- Geometric and Fibonacci terms can grow enormous, so keep the term count modest.
Common mistakes
A frequent confusion is mixing up the position counting. The first term sits at position 1, so the formula for the nth term uses n − 1 steps of growth, not n.
- Remember the first term is position 1, not position 0.
- A geometric ratio of 1 produces a constant sequence, not growth.
- Very large ratios or many terms can overflow into non-finite values.
Formula
Arithmetic: aₙ = a₁ + (n−1)d; Geometric: aₙ = a₁·rⁿ⁻¹; Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂