In This Guide
The Main Difference
Arithmetic Sequence
Each term is found by adding the same constant difference to the previous term. The pattern is additive and changes linearly.
Geometric Sequence
Each term is found by multiplying the previous term by the same constant ratio. The pattern is multiplicative and changes exponentially.
| Feature | Arithmetic | Geometric |
|---|---|---|
| Repeated operation | Add or subtract the same number | Multiply or divide by the same nonzero number |
| Constant | Common difference d | Common ratio r |
| nth term | aₙ = a₁ + (n - 1)d | aₙ = a₁rⁿ⁻¹ |
| Graph of term number vs term value | Points lie on a line | Points follow an exponential pattern |
| Typical context | Fixed amount added each period | Fixed percentage or factor each period |
Fastest identification test
Subtract consecutive terms. If the differences are constant, the sequence is arithmetic. Divide consecutive nonzero terms. If the ratios are constant, the sequence is geometric.
What Is an Arithmetic Sequence?
An arithmetic sequence has a constant difference between consecutive terms. The common difference is found by subtracting one term from the next:
Example
Consider the sequence:
The consecutive differences are:
The common difference is 5, so the sequence is arithmetic.
Decreasing arithmetic sequences
The common difference may be negative:
What Is a Geometric Sequence?
A geometric sequence has a constant ratio between consecutive nonzero terms. Divide a term by the previous term:
Example
The consecutive ratios are:
The common ratio is 2, so the sequence is geometric.
Geometric decay
When the absolute value of the ratio is between 0 and 1, term magnitudes decrease:
Alternating geometric sequences
A negative ratio causes signs to alternate:
The common ratio is -2.
How to Identify a Sequence from a Table
When terms are listed by position, compare consecutive term values. Do not compare terms separated by different numbers of steps unless you adjust for the interval.
Arithmetic table
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| aₙ | 7 | 11 | 15 | 19 | 23 |
| Difference | - | +4 | +4 | +4 | +4 |
Geometric table
| n | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| aₙ | 2 | 6 | 18 | 54 | 162 |
| Ratio | - | 3 | 3 | 3 | 3 |
Zero terms
Ratio tests require division, so a zero term can make direct ratio checking impossible. Use the pattern, formula, or surrounding terms carefully.
Finding the nth Term
Arithmetic nth-term formula
The factor n - 1 appears because moving from the first term to the nth term requires n - 1 equal steps.
Example 1: Arithmetic nth term
The first term is 8 and the common difference is 5.
Answer
The 25th term is 128.
Geometric nth-term formula
Example 2: Geometric nth term
The first term is 5 and the common ratio is 3.
Answer
The 8th term is 10,935.
Sum of an Arithmetic Series
A sequence is a list of terms. A series is the sum of those terms. The sum of the first n terms of an arithmetic sequence can be found using the first and last terms:
An equivalent form is:
Example 3: Arithmetic series sum
First find the 20th term:
Then use the sum formula:
Answer
The sum of the first 20 terms is 820.
Sum of a Finite Geometric Series
For a geometric sequence with ratio r not equal to 1, the sum of the first n terms is:
An equivalent version reverses both subtraction orders:
Example 4: Finite geometric sum
The first term is 2, the ratio is 3, and n = 6.
Answer
The sum of the first 6 terms is 728.
Infinite Geometric Series
An infinite geometric series has a finite sum only when the absolute value of the common ratio is less than 1.
When this condition is satisfied:
Example 5: Infinite geometric series
The ratio is 1/2, so the series converges.
Answer
The infinite series has sum 24.
Divergent infinite series
When |r| is greater than or equal to 1, the terms do not approach zero in the required way, so the infinite geometric series does not have a finite sum.
Real-World Applications
Arithmetic growth: fixed amount
A person saves $50 every month in addition to an initial $200. The account balance, ignoring interest, follows an arithmetic pattern:
Geometric growth: fixed percentage
An investment grows by 6% each year. Each year's balance is multiplied by 1.06:
Geometric decay: depreciation
A machine loses 15% of its value each year. It retains 85% of its value:
Amount versus percent
A fixed amount added repeatedly creates an arithmetic sequence. A fixed percentage applied repeatedly creates a geometric sequence.
Recursive Formulas
A recursive formula defines each term using the previous term.
Arithmetic recursive form
Geometric recursive form
Recursive formulas require a starting value. Explicit formulas allow you to calculate a distant term directly without finding all previous terms.
Common Sequence and Series Mistakes
Using n instead of n - 1
The nth-term formulas count the number of steps from the first term, which is n - 1.
Confusing difference and ratio
Arithmetic uses subtraction to test the pattern; geometric uses division.
Mixing a term formula with a sum formula
aₙ finds one term. Sₙ finds the total of several terms.
Ignoring a negative ratio
A negative geometric ratio alternates signs and must remain negative in powers.
Using the infinite-sum formula without checking convergence
The formula works only when the absolute value of r is less than 1.
Assuming every percent problem is geometric
A one-time percent calculation is not a sequence. Repeated percent change creates a geometric pattern.
Rounding terms too early
Repeated rounding can create noticeable error in later terms and sums.
Forgetting the first term in a recursive definition
A recursion rule cannot generate a unique sequence without an initial value.
A Fast Sequence Checklist
- Are consecutive differences constant?
- Are consecutive nonzero ratios constant?
- What are the first term, common difference, or common ratio?
- Does the problem ask for one term or a sum?
- Should you use an explicit or recursive formula?
- For a geometric sequence, is the ratio negative, fractional, or greater than 1?
- For an infinite series, is |r| less than 1?
- Does the context describe a fixed amount or a fixed percentage?
How Arithmetic and Geometric Patterns Connect to Functions
When the term number n is treated as the input, an arithmetic sequence is the discrete version of a linear function. Its outputs change by a constant amount.
A geometric sequence is the discrete version of an exponential function. Its outputs change by a constant factor. This is why arithmetic sequences appear as equally spaced points on a line, while geometric sequences follow growth, decay, or alternating exponential patterns.
Practice Sequences and Series
Practice arithmetic sequences, geometric sequences, nth terms, finite sums, infinite series, recursive rules, and mixed exam-style questions.