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# Arithmetic Progressions — nth Term, Sum, Examples

## TL;DR

An arithmetic progression (AP) is a sequence where each term increases by a fixed amount called the common difference d, like 2, 5, 8, 11,… This article covers the definition, the nth-term formula an = a + (n−1)d, both sum formulas, a quick derivation, six worked examples, and the mistakes students make most.

### What is an Arithmetic Progression?

An **arithmetic progression** is a sequence of numbers in which the difference between any two consecutive terms is constant. That constant is the **common difference**, written d. Each new term is the one before it plus d.

A general AP looks like this:

a, a + d, a + 2d, a + 3d, …

where a is the first term. Note that d can be negative (a decreasing AP) or zero (a constant sequence). It just has to be the same every step.

### The Arithmetic Progression Formulas

- **nth term (general term):**
  
  an = a + (n−1)d

- **Sum of the first n terms (when you know a and d):**
  
  Sn = n/2 [2a + (n−1)d]

- **Sum of the first n terms (when you know the first and last terms):**
  
  Sn = n/2(a + l)

The two sum formulas are based on the same idea, just formulated differently.

### Where the nth-term formula comes from

It is based on repeated addition: the first term stays the same, the second term is the first term plus d, the nth term accumulates d (n−1) times.

### Where the sum formula comes from (Gauss's trick)

Write the sum in reverse; pair terms, and you can derive the sum formula.

## Examples of Arithmetic Progressions

### Example 1
**Find the common difference of 5, 8, 11, 14,…**

Subtract consecutive terms: 8−5=3 and 11−8=3.

Final answer: d = 3.

### Example 2
**Find the 10th term of 7, 12, 17, 22,…**

Correctly applying the formula:

a10 = a + (n−1)d = 7 + (10−1)(5) = 52.

Final answer: 52.

### Example 3
**Which term of 3, 8, 13, 18,… equals 98?**

Solve for n:

98 = 3 + (n−1)(5) 
\n95 = (n−1)(5) 
\nn = 20.

Final answer: the 20th term.

### Example 4
**Find the sum of the first 20 terms of 4, 9, 14, 19,…**

Using the sum formula:

S20 = 20/2 [2(4) + (20−1)(5)] = 1030.

Final answer: 1030.

### Example 5
**Add all the integers from 1 to 100.**

This sequence is an AP:

S100 = 100/2(1 + 100) = 5050.

Final answer: 5050.

### Example 6
**A starting salary is 30,000, rising by 2,000 each year. What is the total earned over 10 years?**

Using the sum formula:

S10 = 10/2 [2(30000) + (10−1)(2000)] = 390000.

Final answer: 390,000 earned over the decade.

## Where Arithmetic Progressions Show Up

The constant-step pattern is seen in:

- **Salaries and savings**
- **Loan repayment and depreciation**
- **Seating, stacking, and scheduling**

The pairing trick that powers the sum formula is usually credited to **Carl Friedrich Gauss**.

## Common Mistakes With Arithmetic Progressions

### Mistake 1: Using n instead of (n−1) in the nth-term formula.

### Mistake 2: Getting the common difference backwards.

### Mistake 3: Picking the wrong sum formula.

## Conclusion

An **arithmetic progression** adds a constant **common difference** at every step. The nth term and the sum formulas are crucial to solving AP problems efficiently.