Arithmetic Sequence
Description
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant is called the **common difference ($d$)**.
The formula $a_n = a_1 + (n-1)d$ allows you to jump directly to any term in the sequence without listing all the previous ones. * $a_1$ is the starting point. * $d$ is the step size. * $n-1$ is the number of steps you take from the start.
Why $n-1$? Because to get to the first term ($n=1$), you take zero steps. To get to the second term ($n=2$), you take one step. To get to the 100th term, you take 99 steps.
History & Origins
Arithmetic progressions have been studied for thousands of years. Ancient Egypt (c. 1550 BC): The Rhind Mathematical Papyrus contains a famous problem: "Divide 100 loaves of bread among 5 men such that the shares are in an arithmetic progression and one-seventh of the sum of the three largest shares is equal to the sum of the two smallest." This shows a sophisticated understanding of linear sequences. Babylonians: Used arithmetic sequences to calculate the motion of planets and the length of daylight throughout the year.
Derivation by Iteration
We can see the pattern by writing out the first few terms.
Term 1: $a_1$
Term 2: $a_2 = a_1 + d$
Term 3: $a_3 = a_2 + d = (a_1 + d) + d = a_1 + 2d$
Term 4: $a_4 = a_3 + d = (a_1 + 2d) + d = a_1 + 3d$
Pattern: The coefficient of $d$ is always one less than the term number ($n$).
Generalizing for the nth term: $a_n = a_1 + (n-1)d$.
Variables
| Symbol | Meaning |
|---|---|
aₙ | The nth term (the value you want to find) |
a₁ | First term (Starting value) |
n | Position of the term (e.g., 10 for 10th term) |
d | Common difference (Step size) |
Examples
Basic Calculation
Problem: Find the 50th term of the sequence 3, 7, 11, 15...
Solution:
Taxi Fare
Problem: A taxi charges a $5 base fee plus $2 per mile. What is the cost for a 10-mile ride?
Solution: $25
- This is an arithmetic sequence where $a_1$ corresponds to mile 0 (base fee) or we can frame it as: Cost = Base + (miles $\times$ rate).
- Using formula terms: Base ($a_1$) is the start? Technically, at mile 1, cost is $5+2=7$.
- Let's define $a_n$ as cost at mile $n$.
- Mile 1 ($n=1$): $5 + 2(1) = 7$. Wait, the formula $a_n = a_1 + (n-1)d$ assumes we start adding at step 2. This is a common confusion.
- Better approach: Think of $a_0 = 5$ (base). Then $a_n = 5 + 2n$.
- Using standard formula with $a_1=7$ (mile 1) and $d=2$: Cost for 10 miles ($n=10$) is $a_{10} = 7 + (10-1)2 = 7 + 18 = 25$.
- Result: $25.
Depreciation
Problem: A machine is bought for $20,000 and depreciates by $1,500 each year. What is its value after 8 years?
Solution: $8,000
- Starting value (Year 0): 20,000.
- Value at Year 1 ($a_1$): $20,000 - 1,500 = 18,500$.
- Common difference $d = -1,500$.
- We want value after 8 years ($a_8$).
- Formula: $a_8 = a_1 + (8-1)d$.
- $a_8 = 18,500 + 7(-1,500) = 18,500 - 10,500 = 8,000$.
Common Mistakes
Using n instead of n-1
The most common error is writing $a_n = a_1 + nd$. This counts one too many steps. Remember you start *at* the first term, so you take zero steps for term 1.
Confusing sequence with series
This formula ($a_n$) finds a specific *value* in the list. If you want the *sum* of the list, use the Arithmetic Series formula ($S_n$).
Real-World Applications
Finance: Simple Interest
Simple interest grows arithmetically. If you invest money at a fixed interest rate (not compounding), your balance increases by the same amount every year. The balance at year $n$ follows the arithmetic sequence formula.
Computer Science
Memory addresses for arrays are calculated using an arithmetic sequence. If the first element is at address 1000 and each element is 4 bytes, the $n$-th element is at address $1000 + (n-1)4$.
Frequently Asked Questions
Can d be negative?
Yes! If the sequence is decreasing (e.g., 10, 8, 6...), $d$ is negative (-2).
What is the difference between Arithmetic and Geometric?
Arithmetic adds (constant difference). Geometric multiplies (constant ratio).