9.1 Arithmetic and Geometric Sequences

Arithmetic and geometric sequences describe how numbers grow or shrink in predictable, repeating patterns. Arithmetic sequences grow by adding the same amount each time; geometric sequences grow by multiplying by the same amount each time.

These two types form the foundation for understanding series, financial models (like loan payments and compound interest), and many science applications. Once you're comfortable identifying and writing formulas for these sequences, the rest of this unit builds directly on top of them.

Arithmetic vs Geometric Sequences

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Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms. That difference is called the common difference, written as $d$. You get each new term by adding $d$ to the previous one.

• $n$th term formula: $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference
• Sum of the first $n$ terms ($S_n$):
  • $S_n = \frac{n}{2}(2a_1 + (n-1)d)$
  • $S_n = \frac{n}{2}(a_1 + a_n)$

The second sum formula is often faster when you already know the last term $a_n$. It's just the average of the first and last terms, multiplied by the number of terms.

Example: In the sequence $4, 7, 10, 13, ...$, the common difference is $d = 3$. The 10th term is $a_{10} = 4 + (10-1)(3) = 31$. The sum of the first 10 terms is $S_{10} = \frac{10}{2}(4 + 31) = 175$.

Geometric Sequences

A geometric sequence has a constant ratio between consecutive terms. That ratio is called the common ratio, written as $r$. You get each new term by multiplying the previous one by $r$.

• $n$th term formula: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio
• Sum of the first $n$ terms ($S_n$):
  • $S_n = \frac{a_1(1 - r^n)}{1 - r}$ for $r \neq 1$
  • $S_n = n \cdot a_1$ for $r = 1$ (since every term is the same)

Notice that the sum formula is undefined when $r = 1$ because you'd be dividing by zero, which is why the $r = 1$ case needs its own formula.

Example: In the sequence $3, 12, 48, 192, ...$, the common ratio is $r = 4$. The 6th term is $a_6 = 3 \cdot 4^{5} = 3072$. The sum of the first 6 terms is $S_6 = \frac{3(1 - 4^6)}{1 - 4} = \frac{3(1 - 4096)}{-3} = 4095$.

Properties of Sequences

Identifying Sequence Types

To classify a sequence, test both operations:

1. Check for arithmetic: Subtract consecutive terms ($a_2 - a_1$, $a_3 - a_2$, etc.). If all the differences are equal, it's arithmetic.

2. Check for geometric: Divide consecutive terms ($\frac{a_2}{a_1}$, $\frac{a_3}{a_2}$, etc.). If all the ratios are equal, it's geometric.

3. If neither the differences nor the ratios are constant, the sequence is neither arithmetic nor geometric.

Watch out: A sequence like $5, 5, 5, 5, ...$ is technically both arithmetic ($d = 0$) and geometric ($r = 1$). Also, a geometric sequence can never contain a zero term, because you can't divide by zero to find a ratio, and no fixed ratio times zero produces a nonzero next term.

Sequence Notation and Terminology

Sequences use subscript notation: $a_1, a_2, a_3, ..., a_n$, where $a_n$ represents the term at position $n$.

• $a_1$ is the first term, $a_2$ is the second, and so on
• The variable $n$ represents the position (or index) of a term

For example, in the sequence $3, 7, 11, 15, ...$, we'd say $a_1 = 3$, $a_2 = 7$, and $a_3 = 11$.

Finding Common Difference or Ratio

Calculating Common Difference

The most direct method: subtract any term from the next one.

$d = a_{n+1} - a_n$

If you're given non-consecutive terms, rearrange the $n$th term formula:

$d = \frac{a_n - a_1}{n - 1}$

Example: If $a_1 = 2$ and $a_5 = 14$, then $d = \frac{14 - 2}{5 - 1} = \frac{12}{4} = 3$.

Calculating Common Ratio

Divide any term by the previous one:

$r = \frac{a_{n+1}}{a_n}$

If you're given non-consecutive terms, rearrange the $n$th term formula:

$r = \sqrt[n-1]{\frac{a_n}{a_1}}$

Example: If $a_1 = 3$ and $a_4 = 81$, then $r = \sqrt[3]{\frac{81}{3}} = \sqrt[3]{27} = 3$.

Be careful with even roots here. If $a_1 = 2$ and $a_3 = 18$, then $r = \sqrt{\frac{18}{2}} = \sqrt{9} = \pm 3$. Both $r = 3$ (giving $2, 6, 18$) and $r = -3$ (giving $2, -6, 18$) are valid unless the problem gives you more information to rule one out.

Explicit and Recursive Formulas

These are two different ways to describe the same sequence. The difference is practical: explicit formulas let you jump straight to any term, while recursive formulas define each term based on the one before it.

Explicit Formulas

An explicit formula gives you the $n$th term directly from $n$, with no need to know previous terms.

• Arithmetic: $a_n = a_1 + (n-1)d$
  • For the sequence $5, 8, 11, 14, ...$: $a_n = 5 + (n-1)(3)$, which simplifies to $a_n = 3n + 2$
• Geometric: $a_n = a_1 \cdot r^{n-1}$
  • For the sequence $2, 6, 18, 54, ...$: $a_n = 2 \cdot 3^{n-1}$

Explicit formulas are more useful when you need, say, the 50th term. You'd rather plug in $n = 50$ once than apply a recursive rule 49 times.

Recursive Formulas

A recursive formula defines each term using the previous term. You always need to state the first term alongside the rule.

• Arithmetic: $a_1 = \text{(first term)}, \quad a_n = a_{n-1} + d$
  • For the sequence $5, 8, 11, 14, ...$: $a_1 = 5, \quad a_n = a_{n-1} + 3$
• Geometric: $a_1 = \text{(first term)}, \quad a_n = a_{n-1} \cdot r$
  • For the sequence $2, 6, 18, 54, ...$: $a_1 = 2, \quad a_n = a_{n-1} \cdot 3$

A common mistake is writing a recursive formula without specifying $a_1$. Without that starting value, the formula doesn't define a unique sequence.