Terms in a Sequence: The Building Blocks of Patterns
What Exactly is a Term?
Let's start with a simple idea: a list. When you make a grocery list, you write items in an order. A sequence in math is similar, but instead of items, it's a list of numbers arranged in a specific order. Each number in that list has a special name: it is a term of the sequence.
For example, look at this sequence of even numbers: 2, 4, 6, 8, 10, .... Here, the number 2 is the first term. The number 4 is the second term. The number 6 is the third term, and so on. The three dots "..." are called an ellipsis and they mean the sequence continues following the same pattern.
Naming Terms: The Power of Subscripts
To talk about terms clearly, especially in longer sequences, we use a labeling system. We give each term a "name" based on its position. We use a letter (often $a$, $b$, or $t$) followed by a small number called a subscript.
For a sequence, we usually denote the first term as $a_1$ (read as "a sub one"), the second term as $a_2$, the third term as $a_3$, and in general, the term in the $n$-th position as $a_n$. Here, $n$ represents the term number or position.
Let's see this in action with the sequence of odd numbers: 1, 3, 5, 7, 9, ...
- $a_1 = 1$ (The first term is 1)
- $a_2 = 3$ (The second term is 3)
- $a_3 = 5$
- $a_4 = 7$
- $a_5 = 9$
This notation is incredibly powerful. If someone asks, "What is the 100th odd number?" we don't have to write out all 99 numbers before it. We can use a rule (a formula) involving $a_n$ to find it directly.
Finding Patterns: Arithmetic and Geometric Sequences
Sequences often follow a rule. Two of the most common and important types are arithmetic and geometric sequences. The relationship between their terms defines them.
In an arithmetic sequence, each term is found by adding or subtracting a fixed number to the previous term. This fixed number is called the common difference, denoted by $d$.
For example, in the sequence 5, 8, 11, 14, 17, ..., the common difference is $d = 3$ because $8 - 5 = 3$, $11 - 8 = 3$, and so on. We can say:
- $a_1 = 5$
- $a_2 = a_1 + d = 5 + 3 = 8$
- $a_3 = a_2 + d = 8 + 3 = 11$
- In general, $a_n = a_1 + (n-1)d$
In a geometric sequence, each term is found by multiplying or dividing the previous term by a fixed number. This fixed number is called the common ratio, denoted by $r$.
For example, in the sequence 2, 6, 18, 54, 162, ..., the common ratio is $r = 3$ because $6 / 2 = 3$, $18 / 6 = 3$, and so on. We can say:
- $a_1 = 2$
- $a_2 = a_1 * r = 2 * 3 = 6$
- $a_3 = a_2 * r = 6 * 3 = 18$
- In general, $a_n = a_1 * r^{(n-1)}$
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Rule to get next term | Add a constant ($d$) | Multiply by a constant ($r$) |
| Key Constant | Common Difference | Common Ratio |
| General Formula for $a_n$ | $a_n = a_1 + (n-1)d$ | $a_n = a_1 * r^{(n-1)}$ |
| Simple Example | 4, 7, 10, 13, ... ($d=3$) | 5, 15, 45, 135, ... ($r=3$) |
| Graph Shape | A straight line when plotting term number vs. term value | An exponential curve when plotting term number vs. term value |
Sequences in Action: From Saving Money to Musical Beats
Terms of sequences are everywhere once you know how to look for them. They help us model and understand patterns in the real world.
Example 1: Saving Money (Arithmetic Sequence)
Imagine you decide to save money. In the first week, you save $5. Each week after, you save an extra $2 more than the previous week. Your weekly savings form a sequence:
- Week 1 (Term $a_1$): $5$
- Week 2 (Term $a_2$): $7$
- Week 3 (Term $a_3$): $9$
- Week 4 (Term $a_4$): $11$
This is an arithmetic sequence with $a_1 = 5$ and $d = 2$. To find how much you save in the 10th week ($a_{10}$), use the formula: $a_{10} = 5 + (10-1)*2 = 5 + 18 = 23$. So, in week 10, you save $23.
Example 2: Cell Division (Geometric Sequence)
A single bacterium splits into two every hour. Starting with one bacterium, the population each hour forms a sequence:
- Hour 0 (Start, Term $a_1$): 1
- Hour 1 (Term $a_2$): 2
- Hour 2 (Term $a_3$): 4
- Hour 3 (Term $a_4$): 8
This is a geometric sequence with $a_1 = 1$ and $r = 2$. The general term is $a_n = 1 * 2^{(n-1)}$. After 10 hours ($a_{11}$ if we start at hour 0 as $a_1$), the population would be $2^{(10)} = 1024$ bacteria!
Example 3: Music and Rhythm
In music, a common rhythmic pattern is the "doubling" pattern of notes: a whole note, half note, quarter note, eighth note. The duration of each note relative to the whole note forms a sequence: 1, 1/2, 1/4, 1/8, 1/16, .... This is a geometric sequence where each term is multiplied by $r = 1/2$ to get the next.
Important Questions
Yes, absolutely. Terms can be any number: positive, negative, zero, or even fractions. For example, the sequence defined by $a_n = -3n$ has terms: $a_1 = -3$, $a_2 = -6$, $a_3 = -9$, etc. The sequence $5, 0, -5, -10, ...$ is arithmetic with $d = -5$.
This is a crucial distinction. A sequence is an ordered list of numbers (terms). A series is the sum of the terms of a sequence. For the sequence $1, 2, 3, 4$, the corresponding series (if we add all terms) is $1 + 2 + 3 + 4 = 10$. The sequence focuses on the individual numbers; the series focuses on their total.
You need to figure out the pattern first. If you suspect it's arithmetic, find the common difference $d$ by subtracting one known term from another. For geometric, find the common ratio $r$ by dividing one term by the previous term. Then use the general formula to find the term you need.
Example: In an arithmetic sequence, $a_5 = 20$ and $a_9 = 36$. Find $a_1$ and $a_{15}$.
From $a_5$ to $a_9$ is 4 steps. The total change is $36 - 20 = 16$. So, $d = 16 / 4 = 4$.
Since $a_5 = a_1 + (5-1)d$, we have $20 = a_1 + 4*4$, so $a_1 = 20 - 16 = 4$.
Now, $a_{15} = a_1 + (15-1)d = 4 + 14*4 = 4 + 56 = 60$.
Understanding the concept of a term is the first and most important step in exploring the fascinating world of sequences. Each term is a unique piece in an ordered puzzle, defined by its position and its relationship to the terms around it. Whether it's the steady addition in an arithmetic sequence or the rapid multiplication in a geometric one, these patterns—built from individual terms—allow us to describe everything from simple weekly savings to the explosive growth of cells. By mastering the notation (like $a_n$) and the formulas for common sequences, you gain a powerful tool for recognizing patterns, solving problems, and making predictions in math and in life.
Footnote
1 Ellipsis (...): A punctuation mark of three dots used to indicate the continuation of a pattern or list, meaning "and so on."
2 Subscript: A small letter or number written slightly below and to the right of a normal letter or number, used in mathematics and science for labeling (e.g., $a_1$).
3 Common Difference (d): The constant amount added (or subtracted) to each term in an arithmetic sequence to get the next term.
4 Common Ratio (r): The constant amount multiplied by (or used to divide) each term in a geometric sequence to get the next term.
5 Series: The sum of the terms of a sequence. For example, for the sequence 2, 4, 6, the corresponding series is 2 + 4 + 6 = 12.