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Arithmetic & Geometric
Sequences as Functions

Master recursive and explicit formulas for sequences, learn to translate between forms, and model real-world situations with precision.

Algebra 2

What Is a Sequence?

A sequence is an ordered list of numbers, where each number is called a term. We write the \(n\)th term as \(a_n\). A sequence is really a function whose domain is the positive integers:

Finite
\(3, 7, 11, 15\): a fixed number of terms.
Infinite
\(2, 4, 8, 16, \ldots\): the pattern continues forever.

Arithmetic Sequences: Constant Difference

An arithmetic sequence adds (or subtracts) the same value, called the common difference \(d\), to get from one term to the next.

3 7 11 15
\(+4\)
\(+4\)
\(+4\)
 \(a_1\) \(a_2\) \(a_3\) \(a_4\)

Geometric Sequences: Constant Ratio

A geometric sequence multiplies by the same value, called the common ratio \(r\), to get from one term to the next.

2 6 18 54
\(\times 3\)
\(\times 3\)
\(\times 3\)
 \(a_1\) \(a_2\) \(a_3\) \(a_4\)

Why Treat Sequences as Functions?

Viewing sequences as functions unlocks powerful tools: you can graph them, find any term directly, and connect them to the broader world of linear and exponential models.

Predict
Jump to the 100th term without computing all 99 before it.
Graph
Plot \((n, a_n)\) to visualize linear vs. exponential growth.
Model
Represent salary raises, population growth, loan payments, and more.

Explicit Formulas: Direct Access

An explicit formula computes any term \(a_n\) directly from the term number \(n\), without needing previous terms.

Arithmetic
$$a_n = a_1 + (n - 1)d$$
Geometric
$$a_n = a_1 \cdot r^{\,n-1}$$

Recursive Formulas: Step by Step

A recursive formula defines each term using the previous term. It always needs an initial condition (the first term).

Arithmetic
$$a_1 = \text{value},\quad a_n = a_{n-1} + d$$
Geometric
$$a_1 = \text{value},\quad a_n = a_{n-1} \cdot r$$

Explicit vs. Recursive: Side by Side

Explicit Recursive
Needs \(a_1\), \(d\) or \(r\), and \(n\) \(a_1\) and the previous term
Strength Jump to any term instantly Naturally describes the pattern
Weakness Must identify \(d\) or \(r\) first Slow for large \(n\)
Analogy GPS coordinates Turn-by-turn directions

Graphing: Linear vs. Exponential

n (term number)
\(a_n\)
 0 5 10 15 20 25 1 2 3 4 5 Arithmetic Geometric

How to Write Both Formulas

1
Identify the type. Check if consecutive terms have a common difference (arithmetic) or a common ratio (geometric).
2
Find \(a_1\), \(d\), or \(r\). Compute \(d = a_2 - a_1\) or \(r = a_2 / a_1\).
3
Write the recursive formula. State \(a_1\) and the rule: \(a_n = a_{n-1} + d\) or \(a_n = a_{n-1} \cdot r\).
4
Write the explicit formula. Substitute into \(a_n = a_1 + (n-1)d\) or \(a_n = a_1 \cdot r^{n-1}\).
5
Verify. Plug in a known term number and confirm the formula gives the correct value.

Example 1: Arithmetic Sequence

Write both formulas for the sequence: \(5,\; 11,\; 17,\; 23,\; 29,\; \ldots\)

1
Type: \(11 - 5 = 6\), \(17 - 11 = 6\). Common difference, so arithmetic.
2
Parameters: \(a_1 = 5\), \(d = 6\).
3
Recursive: \(a_1 = 5,\quad a_n = a_{n-1} + 6\)
4
Explicit: \(a_n = 5 + (n - 1)(6) = 6n - 1\)

Example 2: Geometric Sequence

Write both formulas for the sequence: \(4,\; 12,\; 36,\; 108,\; \ldots\)

1
Type: \(12 / 4 = 3\), \(36 / 12 = 3\). Common ratio, so geometric.
2
Parameters: \(a_1 = 4\), \(r = 3\).
3
Recursive: \(a_1 = 4,\quad a_n = a_{n-1} \cdot 3\)
4
Explicit: \(a_n = 4 \cdot 3^{\,n-1}\)

Example 3: Translating Between Forms

Given the recursive formula \(a_1 = 100,\; a_n = a_{n-1} \cdot 0.8\), write the explicit formula and find \(a_6\).

1
Identify: Multiplying by \(0.8\) means geometric with \(r = 0.8\).
2
Explicit formula: \(a_n = 100 \cdot (0.8)^{n-1}\)
3
Find \(a_6\): \(a_6 = 100 \cdot (0.8)^5 = 100 \cdot 0.32768 = 32.768\)

Example 4: Modeling a Situation

A gym membership costs $40 to join, then $25 per month. Write explicit and recursive formulas for the total cost after \(n\) months.

1
Sequence: \(65,\; 90,\; 115,\; 140,\; \ldots\) (after month 1, 2, 3, 4)
2
Type: Arithmetic with \(a_1 = 65\), \(d = 25\).
3
Recursive: \(a_1 = 65,\quad a_n = a_{n-1} + 25\)
4
Explicit: \(a_n = 65 + (n-1)(25) = 25n + 40\)

Common Mistakes to Avoid

✕ Wrong Exponent
Writing \(a_n = a_1 \cdot r^n\) instead of \(r^{n-1}\). The exponent is \(n - 1\) because the first term has zero multiplications.
✕ Mixing Up \(d\) and \(r\)
Subtracting terms when you should divide (or vice versa). Always ask: is the pattern adding or multiplying?
✕ Forgetting \(a_1\)
A recursive formula is incomplete without stating the initial condition \(a_1\). Without it, the sequence is undefined.

Summary Table

Arithmetic Geometric
Pattern Add \(d\) Multiply by \(r\)
Recursive \(a_n = a_{n-1} + d\) \(a_n = a_{n-1} \cdot r\)
Explicit \(a_n = a_1 + (n-1)d\) \(a_n = a_1 \cdot r^{n-1}\)
Graph Shape Linear (straight line) Exponential (curve)
Function Family \(f(x) = mx + b\) \(f(x) = a \cdot b^x\)

Question 1 of 5

What is the explicit formula for the arithmetic sequence \(7,\; 13,\; 19,\; 25,\; \ldots\)?

Question 2 of 5

Which recursive formula matches the sequence \(3,\; 12,\; 48,\; 192,\; \ldots\)?

Question 3 of 5

Given \(a_n = 5 \cdot 2^{n-1}\), what is \(a_7\)?

Question 4 of 5

A bacteria colony starts with 500 cells and triples every hour. This creates the sequence of populations \(500, 1500, 4500, \ldots\). Which explicit formula models this sequence?

Question 5 of 5

Convert the explicit formula \(a_n = -2 + 5(n - 1)\) to recursive form.