12.1 Sequences

Sequences and series give you a framework for understanding how values change based on their position in a pattern. Whether it's a simple list of numbers growing by the same amount each time or a more complex formula-driven pattern, these tools let you analyze, predict, and calculate efficiently.

Sequences and Series

Generation of initial sequence terms

To generate the first several terms of a sequence, you need to identify the rule governing the pattern and then apply it repeatedly.

• Arithmetic sequences have a constant difference between consecutive terms, called the common difference ($d$). You get each new term by adding $d$ to the previous one.
  • Example: 2, 5, 8, 11, ... has a common difference of 3
• Geometric sequences have a constant ratio between consecutive terms, called the common ratio ($r$). You get each new term by multiplying the previous one by $r$.
  • Example: 3, 6, 12, 24, ... has a common ratio of 2
• Other sequences follow their own specific formula or recursive rule.
  • Example: The Fibonacci sequence (0, 1, 1, 2, 3, 5, ...) defines each term as the sum of the two preceding terms

The key first step on any problem is figuring out which type of sequence you're dealing with. Check for a constant difference first, then a constant ratio, then look for another pattern.

Derivation of nth term formula

Instead of listing out every term to find, say, the 50th value, you can derive a formula that jumps straight to any term.

For arithmetic sequences:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

• Example: For the sequence 3, 7, 11, 15, ..., the first term is 3 and the common difference is 4. So $a_n = 3 + (n - 1) \cdot 4$, which simplifies to $a_n = 4n - 1$. The 20th term would be $a_{20} = 4(20) - 1 = 79$.

For geometric sequences:

$a_n = a_1 \cdot r^{n-1}$

where $a_1$ is the first term and $r$ is the common ratio.

• Example: For the sequence 2, 6, 18, 54, ..., the first term is 2 and the common ratio is 3. So $a_n = 2 \cdot 3^{n-1}$. The 6th term would be $a_6 = 2 \cdot 3^5 = 2 \cdot 243 = 486$.

For other sequences, look at the relationship between the term number $n$ and the term value, then write a formula that captures it.

• Example: For the sequence 1, 4, 9, 16, ..., each term equals its position squared: $a_n = n^2$.

A formula like these that gives you the nth term directly (without needing previous terms) is called a closed-form expression.

Application of factorial notation

Factorial notation is a shorthand for multiplying a positive integer by every positive integer below it:

$n! = n \cdot (n-1) \cdot (n-2) \cdot \ldots \cdot 3 \cdot 2 \cdot 1$

• $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$
• $3! = 6$, and $1! = 1$
• $0! = 1$ by definition (this comes up often, so just memorize it)

Factorials appear in sequence formulas, especially later when you get to the Binomial Theorem. Here's a quick example of a sequence that uses them:

• If $a_n = \frac{n!}{2^n}$, find $a_3$:
  1. Calculate the numerator: $3! = 6$
  2. Calculate the denominator: $2^3 = 8$
  3. Divide: $a_3 = \frac{6}{8} = \frac{3}{4}$

Factorials grow extremely fast. By $10! = 3{,}628{,}800$, you're already in the millions. This rapid growth matters when you're comparing factorial-based sequences to exponential ones.

Calculation of partial sums

A partial sum is the sum of a specific number of terms from the beginning of a sequence. The notation $S_n$ means "the sum of the first $n$ terms."

For arithmetic sequences:

$S_n = \frac{n}{2}(a_1 + a_n)$

This is equivalent to: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$

The second form is useful when you don't already know $a_n$.

• Example: Find the sum of the first 10 terms of 2, 5, 8, 11, ...
  1. Identify: $a_1 = 2$, $d = 3$, $n = 10$
  2. Find $a_{10}$: $a_{10} = 2 + (10-1)(3) = 2 + 27 = 29$
  3. Apply the formula: $S_{10} = \frac{10}{2}(2 + 29) = 5 \cdot 31 = 155$

For geometric sequences (when $r \neq 1$):

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

If $r = 1$, every term is the same, so $S_n = a_1 \cdot n$.

• Example: Find the sum of the first 5 terms of 3, 6, 12, 24, ...
  1. Identify: $a_1 = 3$, $r = 2$, $n = 5$
  2. Apply the formula: $S_5 = \frac{3(1 - 2^5)}{1 - 2} = \frac{3(1 - 32)}{-1} = \frac{3(-31)}{-1} = 93$

Expression of series in summation notation

Summation notation (also called sigma notation) is a compact way to write the sum of many terms without listing them all out:

$\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \ldots + a_n$

The variable $i$ is the index. It starts at the bottom value (here, 1) and counts up to the top value ($n$). For each value of $i$, you evaluate the expression and add the results together.

Arithmetic series example: Express 2 + 5 + 8 + 11 + ... + 29 in summation notation.

• The first term is 2, the common difference is 3, and there are 10 terms.
• The general term is $a_i = 2 + (i-1) \cdot 3$
• In summation notation: $\sum_{i=1}^{10} [2 + (i-1) \cdot 3]$

Geometric series example: Express 3 + 6 + 12 + 24 + 48 in summation notation.

• The first term is 3, the common ratio is 2, and there are 5 terms.
• The general term is $a_i = 3 \cdot 2^{i-1}$
• In summation notation: $\sum_{i=1}^{5} 3 \cdot 2^{i-1}$

To read summation notation, just plug in each value of $i$ one at a time and add up the results. That's all it's doing.

Behavior of Infinite Sequences and Series

When a sequence or series extends forever, the big question is: does it settle down to a finite value, or does it keep growing without bound?

• Convergence means the terms of a sequence (or the partial sums of a series) approach a specific finite number as $n$ gets larger and larger. That number is called the limit.
• Divergence means no such finite limit exists. The values either grow without bound, oscillate, or otherwise fail to settle.

For example, the sequence $a_n = \frac{1}{n}$ converges to 0 because the terms get closer and closer to 0 as $n$ increases. But the sequence $a_n = 2n$ diverges because the terms grow forever.

An infinite series is the sum of all terms in an infinite sequence. Some infinite series converge to a finite sum (like an infinite geometric series with $|r| < 1$), while others diverge. You'll work more with infinite geometric series later in this unit.