Arithmetic Sequence Equations- Formulas and Examples

What Is an Arithmetic Sequence?

An arithmetic sequence is a list of numbers where the difference between consecutive terms stays the same. That's it. No tricks, no curves—just a constant gap.

You start with a first term, add a fixed amount each time, and you get the next number. The fixed amount is called the common difference.

Example: 3, 7, 11, 15, 19...

The gap here is always 4. That's your common difference.

The Core Formulas You Need

Finding Any Term (Explicit Formula)

The formula for the nth term of an arithmetic sequence:

a_n = a₁ + (n - 1)d

Where:

• a_n = the term you want (nth term)
• a₁ = the first term
• n = which term number you want
• d = common difference

Finding the Sum of Terms

When you need to add up the first n terms:

S_n = n/2 × (a₁ + a_n)

This is the first formula students learn. But there's a version that doesn't require you to find a_n first:

S_n = n/2 × [2a₁ + (n - 1)d]

Use whichever is faster for your problem.

Real Examples That Actually Make Sense

Example 1: Finding the 50th Term

Sequence: 5, 9, 13, 17...

Step 1: Identify a₁ = 5

Step 2: Find d = 9 - 5 = 4

Step 3: Plug into a_n = a₁ + (n - 1)d

a₅₀ = 5 + (50 - 1) × 4

a₅₀ = 5 + 49 × 4

a₅₀ = 5 + 196

a₅₀ = 201

Example 2: Finding the Sum of First 30 Terms

Sequence: 2, 5, 8, 11... (a₁ = 2, d = 3)

Use S_n = n/2 × [2a₁ + (n - 1)d]

S₃₀ = 30/2 × [2(2) + (30 - 1) × 3]

S₃₀ = 15 × [4 + 29 × 3]

S₃₀ = 15 × [4 + 87]

S₃₀ = 15 × 91

S₃₀ = 1,365

Example 3: Identifying the Formula From a Sequence

Given: 10, 7, 4, 1, -2...

a₁ = 10

d = 7 - 10 = -3

The explicit formula: a_n = 10 + (n - 1)(-3)

Simplified: a_n = 13 - 3n

Negative differences happen. Sequences can go down.

Quick Reference Table

How To Actually Use These Formulas

Step 1: Extract what you know from the problem.

Read once. Find a₁ and d. If you're not given them directly, calculate them from the first two terms.

Step 2: Decide what you need.

Are you finding a single term or summing multiple terms? Pick the right formula and stop second-guessing.

Step 3: Plug in the numbers.

Substitute carefully. This is where people lose points—transposing digits, using wrong values for n, mixing up formulas.

Step 4: Solve and check.

Verify your answer makes sense. If you found a₁₀₀ and it's smaller than a₅₀ with a positive d, something's wrong.

Common Mistakes That Cost You Points

• Using n instead of (n-1): The formula accounts for the first term already. Don't add an extra term.
• Screwing up negative differences: When d is negative, the sequence decreases. The math still works the same way.
• Forgetting to divide by 2 in sum formulas: This is the most common error. The n/2 factor is not optional.
• Misidentifying a₁: The first term is always the first number listed. Don't assume it's 1.

Practice Problems

1. Find the 25th term of: 8, 14, 20, 26...

2. What is the sum of the first 40 terms of: 3, 7, 11, 15...

3. A sequence starts at 100 and decreases by 5 each term. What is the 15th term?

4. Find the sum of the first 20 terms of: 12, 9, 6, 3...

Answers

1. a₁ = 8, d = 6 → a₂₅ = 8 + 24(6) = 152

2. a₁ = 3, d = 4, n = 40 → S₄₀ = 40/2 × [2(3) + 39(4)] = 20 × 162 = 3,240

3. a₁ = 100, d = -5 → a₁₅ = 100 + 14(-5) = 30

4. a₁ = 12, d = -3, n = 20 → S₂₀ = 20/2 × [2(12) + 19(-3)] = 10 × (24 - 57) = 10 × (-33) = -330

When You'll Actually Use This

Arithmetic sequences show up in loan payments (fixed amounts each month), stadium seating (consistent row increases), salary steps (annual raises of set amounts), and scheduling problems.

You're not going to need this every day. But when a problem asks for it, knowing the two core formulas cold will save you time and prevent stupid mistakes.