Welcome to Chapter 10-11 Notes 4!

In this lesson, we dive deeper into the world of sequences and series, specifically focusing on Geometric Sequences. Unlike arithmetic sequences where we add or subtract to get to the next term, geometric sequences are defined by multiplication. Whether you are looking for a specific term deep in a sequence or trying to sum up an infinite list of numbers, this guide will walk you through the essential formulas you need to succeed.

Missed the class? You can catch up by watching the recording below or joining the Microsoft Teams link provided.

Key Topics Covered

  • Identifying Sequences: Distinguishing between Arithmetic (constant difference) and Geometric (constant ratio).
  • The Common Ratio ($r$): Finding $r$ even when dealing with fractions or signed numbers.
  • Explicit Rules: Writing formulas to find any term ($a_n$) in a sequence.
  • Geometric Series: Calculating the sum of the first $n$ terms and the sum of infinite series.

1. Arithmetic vs. Geometric Sequences

The first step in solving sequence problems is identifying the pattern. Remember:

  • Arithmetic Sequences change by a common difference ($d$). Example: $2, 4, 6, 8...$ (adding 2).
  • Geometric Sequences change by a common ratio ($r$). Example: $3, 6, 12, 24...$ (multiplying by 2).

To find the common ratio $r$, divide any term by the previous term:

$$ r = \frac{a_2}{a_1} = \frac{a_3}{a_2} $$

Note on Signed Numbers: If the signs of the terms alternate (e.g., $2, -6, 18, -54$), your common ratio $r$ will be a negative number.

2. The Explicit Rule for Geometric Sequences

Instead of multiplying repeatedly to find the 50th term, we use the explicit formula. To find the $n$-th term of a geometric sequence, use:

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

Where:

  • $a_n$ is the term you are trying to find.
  • $a_1$ is the first term.
  • $r$ is the common ratio.
  • $n$ is the position number of the term.

3. Finding the Sum of a Finite Geometric Series

If you need to add up the first $n$ terms of a geometric sequence (for example, the first 10 terms), we use the finite sum formula:

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

Be careful when entering this into your calculator, especially if $r$ is negative!

4. Sum of an Infinite Geometric Series

Can you add up a list of numbers that goes on forever? Yes, but only if the terms are getting smaller. This happens when the absolute value of the ratio is less than 1 ($|r| < 1$). If $|r| \geq 1$, the series diverges and has no sum.

The formula for a convergent infinite series is elegantly simple:

$$ S_\infty = \frac{a_1}{1 - r} $$

Watch the Lesson

For detailed examples, including how to find a term given only two other terms of the sequence, be sure to watch the full video breakdown below. Happy studying!