How Do You Find the 100th Term of a Sequence?
Imagine you’re tracking a habit—saving a dollar each day, or the distance a ball bounces after each drop. By day 100, how much will you have saved? How far will the 100th bounce be? The answer lies in understanding sequences and their nth term formulas. Finding the 100th term isn’t about counting manually; it’s about recognizing the underlying pattern and applying the correct mathematical rule. This guide will walk you through the primary methods for arithmetic and geometric sequences, explore other common patterns, and show you how to tackle this problem with confidence, whether you’re a student, a curious learner, or someone applying math in real-world scenarios.
Understanding the Foundation: What Is a Sequence?
A sequence is simply an ordered list of numbers following a specific rule. Each number in the sequence is called a term. The first term is (a_1), the second is (a_2), and so on. The 100th term is denoted (a_{100}). The key to finding any term, especially one far along like the 100th, is identifying the pattern that generates the sequence. The two most fundamental patterns are arithmetic and geometric sequences.
Arithmetic Sequences: The Constant Step
An arithmetic sequence has a constant difference between consecutive terms. This constant is called the common difference, denoted (d).
- Pattern: (a_1, a_1 + d, a_1 + 2d, a_1 + 3d, \dots)
- General Formula (nth Term): (a_n = a_1 + (n-1)d)
This formula is your primary tool. It states that to get to the nth term, you start with the first term and add the common difference ((n-1)) times.
Step-by-Step Example: Consider the sequence: 5, 9, 13, 17, …
- Identify (a_1): The first term is 5.
- Find the common difference (d): Subtract any term from the one after it: (9 - 5 = 4). So, (d = 4).
- Plug into the formula for (n = 100): (a_{100} = 5 + (100 - 1) \times 4) (a_{100} = 5 + 99 \times 4) (a_{100} = 5 + 396) (a_{100} = 401)
Why this works: The 100th term is 99 steps of size (d) away from the first term. You’ve effectively "climbed" 99 steps from your starting point.
Geometric Sequences: The Constant Multiplier
A geometric sequence has a constant ratio between consecutive terms. This constant is the common ratio, denoted (r).
- Pattern: (a_1, a_1 \times r, a_1 \times r^2, a_1 \times r^3, \dots)
- General Formula (nth Term): (a_n = a_1 \times r^{(n-1)})
Here, you multiply the first term by the common ratio raised to the power of ((n-1)).
Step-by-Step Example: Consider the sequence: 2, 6, 18, 54, …
- Identify (a_1): The first term is 2.
- Find the common ratio (r): Divide any term by the one before it: (6 / 2 = 3). So, (r = 3).
- **Plug into the formula for (n = 100):
