Sequences and Series
A fuller guide to arithmetic and geometric sequences and series, with a focus on pattern recognition, nth-term logic, and deciding whether you need a single term or a cumulative total.
Key formulas
Valid when r is not equal to 1.
First ask what stays constant
An arithmetic sequence changes by adding or subtracting the same amount each step. A geometric sequence changes by multiplying by the same factor each step. That single question often tells you which model you are dealing with before you write any formula.
Series are related but different: a sequence lists terms, while a series adds them. Confusing the two is one of the most common reasons for choosing the wrong calculator.
Nth term versus sum
The nth-term formula tells you the value of a specific position in the sequence. The sum formula tells you the total of several terms up to some limit. If the problem asks for the tenth payment, you need a term. If it asks for the total paid across ten periods, you need a sum.
Keeping those purposes separate stops many errors that look algebraic but are really interpretive. A sequence problem can have correct arithmetic and still answer the wrong question if the term/sum distinction was missed.
Arithmetic patterns
For arithmetic sequences, the common difference is constant. If the first term is a and the difference is d, the nth term is a + (n - 1)d. The finite sum can then be found by averaging the first and last terms and multiplying by the number of terms.
Arithmetic models appear in stepwise growth or decline where the same quantity is added each time, such as repeated fixed payments or equally spaced values in a simple table.
Geometric patterns
For geometric sequences, the common ratio is constant. If the first term is a and the ratio is r, the nth term is a x r^(n - 1). Finite geometric sums depend on whether the ratio is above, below, or equal to 1, because that controls how rapidly the terms grow or shrink.
Geometric models appear in compounding, repeated proportional change, and any setting where each term depends multiplicatively on the previous one rather than by a fixed step.
Worked examples
Example 1: Sequence 5, 8, 11, 14 is arithmetic with difference 3. The tenth term is 5 + 9 x 3 = 32.
Example 2: Sequence 2, 6, 18, 54 is geometric with ratio 3. The fifth term is 2 x 3^4 = 162.
Example 3: The sum of the first five terms of 5, 8, 11, 14, 17 is 55. You can verify it directly or use the arithmetic-series formula.
- Term question -> use nth term.
- Total question -> use series sum.
- Equal difference -> arithmetic.
- Equal ratio -> geometric.
Common mistakes and interpretation
- Using an arithmetic model when the change is actually multiplicative.
- Using the common ratio where the common difference should be used, or vice versa.
- Forgetting that n counts positions starting from 1 in the standard form of the formula.
- Treating a sequence as realistic indefinitely when the real-world process has limits.
Apply the topic straight away.
Arithmetic Sequence nth Term Calculator
Use the Arithmetic Sequence nth Term Calculator for a quick arithmetic sequence nth term result with clear inputs and a readable answer.
Arithmetic Series Sum Calculator
Use the Arithmetic Series Sum Calculator for a quick arithmetic series sum result with clear inputs and a readable answer.
Geometric Sequence nth Term Calculator
Use the Geometric Sequence nth Term Calculator for a quick geometric sequence nth term result with clear inputs and a readable answer.
Geometric Series Sum Calculator
Use the Geometric Series Sum Calculator for a quick geometric series sum result with clear inputs and a readable answer.