The nth Term: Unlocking Patterns in Sequences
What is a Sequence and the nth Term?
A sequence is an ordered list of numbers, like 2, 4, 6, 8, 10, ... Each number in a sequence is called a term. The first number is the 1st term, the second is the 2nd term, and so on. The power of mathematics lies in finding patterns so we don't have to list all terms one by one. This is where the nth term comes in.
The nth term is a formula that uses a term's position (n) to calculate its value. The letter 'n' acts as a placeholder for any position number. For the sequence above, the rule is $2n$. This means:
- For the 1st term ($n=1$), the value is $2 × 1 = 2$.
- For the 2nd term ($n=2$), the value is $2 × 2 = 4$.
- For the 100th term ($n=100$), the value is $2 × 100 = 200$.
With the nth term formula, you can find any term instantly without knowing the previous ones. It's like having a shortcut to any point in the pattern.
Finding the nth Term of an Arithmetic Sequence
An arithmetic sequence is one where the difference between consecutive terms is always the same. This constant difference is called the common difference, denoted by 'd'.
Let's find the nth term for the sequence: 5, 8, 11, 14, 17, ...
Step 1: Find the common difference (d). Subtract one term from the next: $8 - 5 = 3$, $11 - 8 = 3$. So, $d = 3$.
Step 2: The general formula for the nth term of an arithmetic sequence is: $a_n = a_1 + (n-1)d$
Here, $a_n$ is the nth term, $a_1$ is the first term, and $d$ is the common difference.
Step 3: Substitute the values. $a_1 = 5$ and $d = 3$. So the formula becomes: $a_n = 5 + (n-1) × 3$.
Step 4: Simplify the formula. $a_n = 5 + 3n - 3$, which simplifies to $a_n = 3n + 2$.
Let's test it! For the 4th term ($n=4$), $a_4 = 3(4) + 2 = 14$, which matches our sequence.
| Position (n) | Term Value | Check with $a_n = 3n + 2$ |
|---|---|---|
| 1 | 5 | $3(1) + 2 = 5$ ✓ |
| 2 | 8 | $3(2) + 2 = 8$ ✓ |
| 3 | 11 | $3(3) + 2 = 11$ ✓ |
| 10 | ? | $3(10) + 2 = 32$ |
Finding the nth Term of a Geometric Sequence
A geometric sequence is one where each term is found by multiplying the previous term by a constant. This constant is called the common ratio, denoted by 'r'.
Let's find the nth term for the sequence: 2, 6, 18, 54, ...
Step 1: Find the common ratio (r). Divide one term by the previous term: $6 ÷ 2 = 3$, $18 ÷ 6 = 3$. So, $r = 3$.
Step 2: The general formula for the nth term of a geometric sequence is: $a_n = a_1 × r^{(n-1)}$
Here, $a_n$ is the nth term, $a_1$ is the first term, and $r$ is the common ratio.
Step 3: Substitute the values. $a_1 = 2$ and $r = 3$. So the formula becomes: $a_n = 2 × 3^{(n-1)}$.
Let's test it! For the 4th term ($n=4$), $a_4 = 2 × 3^{(3)} = 2 × 27 = 54$, which matches our sequence.
Formula Summary:
- Arithmetic Sequence: $a_n = a_1 + (n-1)d$
- Geometric Sequence: $a_n = a_1 × r^{(n-1)}$
Remember, 'd' is for the common difference (add/subtract), and 'r' is for the common ratio (multiply/divide).
Using the nth Term in Real-World Situations
The nth term isn't just for number puzzles; it helps us model and predict real-life events.
Example 1: Saving Money
Imagine you save $10 in the first week and add $5 each week after. This forms an arithmetic sequence: 10, 15, 20, 25, ... Here, $a_1 = 10$ and $d = 5$. The nth term is $a_n = 10 + (n-1) × 5 = 5n + 5$. To find how much you save in the 15th week ($n=15$), calculate $a_{15} = 5(15) + 5 = 80$. You will save $80 in the 15th week.
Example 2: Cell Division
A type of bacteria doubles every hour. Starting with 1 bacterium, the population forms a geometric sequence: 1, 2, 4, 8, ... Here, $a_1 = 1$ and $r = 2$. The nth term is $a_n = 1 × 2^{(n-1)} = 2^{(n-1)}$. To find the population after 7 hours ($n=7$), calculate $a_7 = 2^{(6)} = 64$. There will be 64 bacteria after 7 hours.
Example 3: Seating Arrangements
An auditorium has 20 seats in the first row, 24 in the second, 28 in the third, and so on. This is arithmetic with $a_1 = 20$ and $d = 4$. The nth term is $a_n = 20 + (n-1) × 4 = 4n + 16$. The 10th row ($n=10$) has $a_{10} = 4(10) + 16 = 56$ seats.
Common Mistakes and Important Questions
Q: What is the difference between 'n' and the value of the nth term?
'n' is the position number (1st, 2nd, 3rd,...). The value of the nth term is the actual number that appears in that position in the sequence. For example, in the sequence 5, 8, 11, 14,... when $n=3$ (the position), the value of the nth term is 11.
Q: Why is it $(n-1)$ in the formulas and not just $n$?
This is a common point of confusion. Think about the first term. For the first term, $n=1$. If we used $n$ in the arithmetic formula, we would add the common difference once, giving us $a_1 + d$. But the first term is just $a_1$; we haven't added the common difference yet because there is no previous term. The $(n-1)$ counts how many times we actually add the common difference to the first term to get to the nth term. For the 1st term, we add it 0 times. For the 2nd term, we add it 1 time. For the nth term, we add it $(n-1)$ times.
Q: What if the sequence is decreasing?
The formulas still work! For an arithmetic sequence, a decreasing pattern means the common difference (d) is a negative number. For example, the sequence 20, 17, 14, 11, ... has $d = -3$. The nth term would be $a_n = 20 + (n-1)(-3) = -3n + 23$. For a geometric sequence, a decreasing pattern means the common ratio (r) is a fraction between 0 and 1. For example, 81, 27, 9, 3, ... has $r = 1/3$.
The nth term is a powerful key that unlocks the patterns hidden within sequences. By understanding how to derive and use the formula $a_n = a_1 + (n-1)d$ for arithmetic sequences and $a_n = a_1 × r^{(n-1)}$ for geometric sequences, you move from simply listing numbers to predicting any term with precision. This skill transforms how you view patterns, turning them from mysterious lists into predictable, manageable relationships. Whether you're calculating future savings, modeling population growth, or just solving a math problem, the nth term formula gives you the power to leap directly to the answer.
Footnote
[1] Arithmetic Sequence: A sequence of numbers in which the difference between any two consecutive terms is constant. This difference is called the common difference.
[2] Geometric Sequence: A sequence of numbers in which the ratio between any two consecutive terms is constant. This ratio is called the common ratio.
[3] Common Difference (d): The constant value added to each term in an arithmetic sequence to get the next term. It is found by subtracting a term from the term that follows it.
[4] Common Ratio (r): The constant value multiplied by each term in a geometric sequence to get the next term. It is found by dividing a term by the previous term.