Patterns and Sequences: Mastering the Nth Term
Learn how to spot patterns and calculate the nth term rule.
👋 Welcome to the world of patterns! In this lesson, you will learn how to spot mathematical sequences, describe how they grow, and unlock the powerful "$n$th term" formula to predict the future (mathematically speaking!).
1. Identifying Patterns 🔍
A sequence is an ordered list of numbers that follow a specific rule. Each number in the sequence is called a term.
To understand a sequence, we look at how to get from one term to the next. There are two main types of sequences you will encounter:
🔢 Arithmetic Sequences
In an arithmetic sequence, the difference between consecutive terms is always the same. We call this the common difference.
Example: $3, 7, 11, 15, [[dots]]$
Rule: Add $4$ each time.
🚀 Geometric Sequences
In a geometric sequence, you multiply or divide by the same number each time to get the next term. We call this the common ratio.
Example: $2, 6, 18, 54, [[dots]]$
Rule: Multiply by $3$ each time.
2. Term-to-Term Rules 📝
The simplest way to describe a sequence is the term-to-term rule. This tells you how to find the next number if you know the current one.
Let’s look at the sequence: $5, 12, 19, 26, [[dots]]$
To find the rule, ask yourself: "How do I get from $5$ to $12$?"
$$12 – 5 = 7$$
Does it work for the next pair? Yes, $19 – 12 = 7$.
So, the term-to-term rule is: Add $7$.
3. The Nth Term Rule ⚡
The term-to-term rule is useful, but what if you need to find the $100$th term? Adding $7$ one hundred times would take forever! This is where the position-to-term rule, or the $n$th term, becomes essential.
In algebra, we use the letter $n$ to represent the position of a number in the sequence.
- $n = 1$ is the 1st term.
- $n = 2$ is the 2nd term.
- $n = 100$ is the 100th term.
How to find the $n$th term for linear sequences
For arithmetic sequences (where you add or subtract the same amount), use this method:
The Formula Strategy
$$T_n = dn + c$$
Where:
- $d$ is the difference between terms.
- $c$ is the zero term (the term that would come before the first term).
Let’s find the $n$th term for: $4, 7, 10, 13, [[dots]]$
Step 1: Find the difference ($d$).
The sequence goes up by $3$ each time ($7 – 4 = 3$). So, the first part of our rule is $3n$.
Step 2: Find the zero term ($c$).
Go backwards from the first term. If we subtract $3$ from the first term ($4$), we get $1$. So, $c = +1$.
Step 3: Combine them.
The $n$th term rule is $3n + 1$.
💡 Check your answer!
If the rule is $3n + 1$, let’s test it for the 3rd term ($n=3$).
$$3 [[times]] 3 + 1 = 10$$
Looking at our sequence ($4, 7, 10, 13$), the 3rd term is indeed $10$. It works!
Practice Problems
Practice finding the rules for these sequences.
1. Find the next two terms in this arithmetic sequence:
$$8, 13, 18, 23, [[dots]]$$
Show Answer
Step 1: Find the difference between the terms.
$$13 - 8 = 5$$
The rule is "add $5$".
Step 2: Apply the rule to the last known term ($23$).
$$23 + 5 = 28$$
$$28 + 5 = 33$$
The next two terms are $28$ and $33$.
2. Find the $n$th term rule for the sequence:
$$5, 9, 13, 17, [[dots]]$$
Show Answer
Step 1: Find the common difference ($d$).
The sequence increases by $4$ each time. So, start with $4n$.
Step 2: Find the zero term ($c$).
Work backwards from the first term ($5$) by subtracting the difference ($4$).
$$5 - 4 = 1$$
So, the adjustment is $+1$.
Answer: The $n$th term is $4n + 1$.
3. Use the $n$th term rule $6n - 2$ to find the 50th term.
Show Answer
Step 1: Substitute $n = 50$ into the formula.
$$6(50) - 2$$
Step 2: Calculate.
$$300 - 2 = 298$$
The 50th term is $298$.
Interactive Quiz
Test your knowledge of patterns and sequences!