Sequences: Arithmetic & Geometric

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Identify whether a sequence is arithmetic or geometric.
  • Use formulas for the nth term and for the common difference/ratio.
  • Recognize how different types of sequences grow (linear vs exponential).
  • Compute missing terms and apply sequence formulas confidently.

Key Ideas

A sequence is an ordered list of numbers that follow a rule.

In this lesson we focus on the two most common types:

  • Arithmetic sequences (add/subtract the same amount each step)
  • Geometric sequences (multiply/divide by the same amount each step)

Arithmetic Sequences

Arithmetic sequences change by adding the same value each time.

  • That fixed value is the common difference, \(d\).

Example:
\[ 5,\; 8,\; 11,\; 14,\; 17,\ldots \quad (d = 3) \]

nth-term formula
If \(a_1\) is the first term:

\[ a_n = a_1 + (n-1)d \]

This represents linear growth.

Geometric Sequences

Geometric sequences change by multiplying by the same value each time.

  • That multiplier is the common ratio, \(r\).

Example:
\[ 2,\; 6,\; 18,\; 54,\ldots \quad (r = 3) \]

nth-term formula
If \(a_1\) is the first term:

\[ a_n = a_1 \cdot r^{\,n-1} \]

This represents exponential growth.

Visualization of arithmetic (linear) vs geometric (exponential) growth.
Important

If the pattern changes by adding/subtracting, it’s arithmetic.
If it changes by multiplying/dividing, it’s geometric.

Common Problem Types

1. Finding the Next Terms

Example (Arithmetic):
Sequence: \(12,\; 7,\; 2,\ldots\)
\(d = -5\)

Next term: \(2 - 5 = -3\)

Example (Geometric):
Sequence: \(4,\; 10,\; 25,\ldots\)
\(r = \frac{10}{4} = 2.5\)

Next term: \(25 \cdot 2.5 = 62.5\)

2. Using the nth-Term Formula (Arithmetic)

Example:
Find \(a_{20}\) for \(a_1 = 7\) and \(d = 4\):

\[ a_{20} = 7 + (20-1)4 = 7 + 76 = 83 \]

3. Using the nth-Term Formula (Geometric)

Example:
Find \(a_6\) for \(a_1 = 3\) and \(r = 2\):

\[ a_6 = 3 \cdot 2^{5} = 96 \]

4. Identifying Type Using Differences or Ratios

  • Constant differences → arithmetic
  • Constant ratios → geometric

Example:
Sequence: \(9,\; 6,\; 4,\; \frac{8}{3},\ldots\)

Ratios:

\[ \frac{6}{9}=\frac{2}{3},\quad \frac{4}{6}=\frac{2}{3} \]

Constant ratio → geometric.

5. Finding a Missing Term

Example:
In an arithmetic sequence, \(a_1 = 10\) and \(a_4 = 4\). Find \(d\).

Use \(a_n = a_1 + (n-1)d\):

\[ 4 = 10 + 3d \] \[ 3d = -6 \] \[ d = -2 \]

Strategies

  • Check differences first, then ratios.
  • Arithmetic → steady, linear growth.
  • Geometric → grows or shrinks rapidly.
  • Use nth-term formulas for distant terms.
  • Visualizing patterns often reveals the type quickly.

Worked Examples

Example 1 (Arithmetic)

Find the 12th term of \(4,\; 9,\; 14,\; 19,\ldots\)

\(d = 5\)

\[ a_{12} = 4 + 11 \cdot 5 = 59 \]

Answer: \(59\)

Example 2 (Geometric)

Find the 7th term of \(a_1 = 5\), \(r = \frac{1}{2}\).

\[ a_7 = 5 \cdot \left(\frac12\right)^6 = \frac{5}{64} \]

Answer: \(\frac{5}{64}\)

Example 3 (Identify Sequence Type)

Sequence: \(3,\; 6,\; 12,\; 24,\ldots\)

Ratios:

\[ 2,\; 2,\; 2 \]

Answer: Geometric, \(r = 2\)

WarningCommon Mistakes
  • Calling a sequence arithmetic after only checking one pair of terms.
  • Using differences for geometric sequences instead of ratios.
  • Forgetting the exponent is \(n-1\) in the geometric nth-term formula.
  • Mixing up the first term \(a_1\) with the term number \(n\).
  • Assuming every growing sequence is arithmetic.

Practice Problems

  1. Identify the sequence type: \(7,\; 11,\; 15,\; 19,\ldots\)
  2. Identify the sequence type: \(100,\; 50,\; 25,\; 12.5,\ldots\)
  3. Arithmetic: \(a_1 = 2\), \(d = -3\). Find \(a_{10}\).
  4. Geometric: \(a_1 = 9\), \(r = \frac{1}{3}\). Find \(a_5\).
  5. In a geometric sequence, \(a_1 = 4\) and \(a_4 = 108\). Find \(r\).

1. Differences → \(+4\) each time → arithmetic
Answer: Arithmetic

2. Ratio → \(\frac12\) each time → geometric
Answer: Geometric, \(r=\frac12\)

3.
\[ a_{10} = 2 + 9(-3) = -25 \]

4.
\[ a_5 = 9 \left(\frac13\right)^4 = \frac19 \]

5.
Use \(a_4 = a_1 r^3\):

\[ 108 = 4r^3 \] \[ r^3 = 27 \] \[ r = 3 \]

Summary

  • Arithmetic sequences add or subtract the same value each step.
  • Geometric sequences multiply or divide by the same value each step.
  • Use the nth-term formulas to reach terms far into the sequence.
  • Checking differences and ratios quickly identifies the sequence type.
  • Arithmetic → constant difference.
  • Geometric → constant ratio.
  • Visualizing the pattern often reveals the rule immediately.