# Common Difference — Arithmetic Progression Definition

The common difference **d** in an arithmetic progression is the fixed value added to each term to get the next:  
d = a_n - a_{n-1}, the same for every consecutive pair. This article covers the definition, three worked examples, the n-th term and sum formulas that use **d**, and how to find **d** when only two non-adjacent terms are known.

## The Number That Makes an Arithmetic Sequence "Arithmetic"

Some sequences grow by multiplying. Others grow by adding the same amount each step. The second kind — arithmetic sequences — are completely characterised by a single number: the common difference.

Once you know the first term and the common difference, you know every term of the sequence forever. The common difference is the "rate" of an arithmetic progression — its slope.

## What the Common Difference Is

The **common difference** **d** of an arithmetic progression (AP) is the constant difference between any term and its predecessor:

d = a_n - a_{n-1} = a_{n+1} - a_n = ... .

This must be the _same_ value for every consecutive pair. If the differences vary, the sequence is not arithmetic.

**Examples.**  
5, 8, 11, 14, 17,… — **d** = 3.  
100, 95, 90, 85,… — **d** = -5.  
2.5, 4.0, 5.5, 7.0,… — **d** = 1.5.  
7, 7, 7, 7,… — **d** = 0 (constant sequence).

> **Quick facts.**  
> - **Formula:** d = a_n - a_{n-1}.  
> - **Sign:** can be positive (increasing AP), negative (decreasing AP), or zero (constant AP).  
> - **Value:** any real number — integer, fraction, decimal, irrational.  
> - **n-th term of AP:** a_n = a_1 + (n - 1)d.  
> - **Sum of first n terms:** S_n = \tfrac{n}{2}(2a_1 + (n-1)d) = \tfrac{n}{2}(a_1 + a_n).  
> - **Two non-adjacent terms:** d = (a_m - a_n) / (m - n).

## How to Find the Common Difference

### **Case 1 — Adjacent terms given**  
Subtract any term from the next.

**Example.** For 4, 9, 14, 19,…:  
d = 9 - 4 = 5. (Confirm: 14 - 9 = 5 ✓.)

### **Case 2 — Two non-adjacent terms given**  
Use d = (a_m - a_n) / (m - n).

**Example.** The 3rd term is 11 and the 7th term is 23.  
d = (23 - 11) / (7 - 3) = 12 / 4 = 3.

### **Case 3 — From the n-th term formula**  
If a_n = 5n + 2, then **d** is the coefficient of **n** in linear form = 5.

## The n-th Term Formula

The general term of an arithmetic progression is  
a_n = a_1 + (n - 1)d.

**Derivation.**  
a_2 = a_1 + d.  
a_3 = a_2 + d = a_1 + 2d.  
By induction, a_n = a_1 + (n - 1)d.

The exponent on **d** is n-1 because the first term involves zero additions: a_1 = a_1 + 0⋅d.

## Three Worked Examples — Quick, Standard, Stretch

**Quick.** Find the common difference of 3, 7, 11, 15, 19,…  
d = 7 - 3 = 4. Confirm: 11 - 7 = 4, 15 - 11 = 4. Constant.  
Final answer: d = 4.

**Standard (Wrong Path First — Where Students Trip Up).**  
The 4th term of an AP is 17 and the 9th term is 32. Find the common difference.  
_The wrong path._ The rusher subtracts: 32 - 17 = 15. They report d = 15.  
_The rescue._ Apply the non-adjacent-terms formula.

d = (a_9 - a_4) / (9 - 4) = (32 - 17) / 5 = 3.  
Final answer: d = 3.

**Stretch.** The first term of an AP is 5 and the sum of the first 10 terms is 185. Find the common difference.

185 = \tfrac{10}{2}(2 × 5 + 9d).  
37 = 10 + 9d.  
9d = 27 ⟹ d = 3.  
Final answer: d = 3.

## Why the Common Difference Matters — Beyond Pattern Recognition

The common difference is the slope of an arithmetic progression. Once you know it, you can compute any term, any sum, any prediction about the sequence.

- **Salary increments.** A starting salary of 40,000 with a yearly increment of 3,000 is an AP.
- **Loan repayments.** Fixed-instalment loans form an AP.
- **Stadium seating.** Rows expanding by a constant number of seats each row form an AP.
- **Counting problems.** Integers divisible by 7 form an AP.
- **Linear motion.** An object moving at constant velocity passes equally-spaced markers at equal time intervals — an AP in position.

## The Tripping Points of Common Difference

### **1. Subtracting non-adjacent terms without averaging.**  
**Don't do this:** Treat a_9 - a_4 as the common difference.  
**The correct way:** d = (a_9 - a_4) / (9 - 4).

### **2. Confusing the common difference with the common ratio.**  
**Don't do this:** Treat any constant pattern as arithmetic.

### **3. Forgetting that d can be zero.**  
**Don't do this:** Restrict **d** to non-zero values.

### **4. Misapplying the index in the n-th term formula.**  
**Don't do this:** Drop -1 in the index.  
**The correct way:** The first term involves zero additions.

## The Mathematicians Who Worked With Arithmetic Progressions

**Aryabhata (476–550 CE, India)** gave the formula for the sum of an arithmetic progression.

**Carl Friedrich Gauss (1777–1855, Germany)** famously summed 1+2+⋯+100 in seconds as a child by recognising the AP sum formula.

**Pierre de Fermat (1607–1665, France)** worked with arithmetic progressions in number theory.

## Conclusion

- The **common difference** **d** is the constant added to each term of an arithmetic progression.
- For adjacent terms, d = a_n - a_{n-1}.
- For non-adjacent terms, d = (a_m - a_n) / (m - n).
- The n-th term is a_n = a_1 + (n - 1)d.
- The most common mistake is subtracting non-adjacent terms without dividing by the index gap.

## Sharpen Your Common Difference — Three Practice Problems

1. Find the common difference of −2, 1, 4, 7, 10,….
2. The 5th term of an AP is 22, and the 12th term is 50. Find the common difference.
3. The first term is 7 and the sum of the first 8 terms is 92. Find the common difference.