Math Lesson

ARITHMETIC SEQUENCE

An arithmetic sequence is a sequence where each term is obtained by adding a constant difference to the previous term!

An arithmetic sequence adds the same number every time. That constant number is called the common difference (d). Once you know the first term and d, you can find any term in the sequence!

a_n = a₁ + (n − 1)d

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is always the same. This constant difference is called d (common difference).

Example: 2, 5, 8, 11, 14, ...
Each term increases by 3, so the common difference is d = 3.

Key Formulas

1. Common Difference:
d = a₂ − a₁

2. nth Term Formula:
a_n = a₁ + (n − 1)d

Where:
a_n = the nth term you want to find
a₁ = the first term
n = the position of the term
d = the common difference

4 Worked Examples

Example 1: Find the common difference of 4, 9, 14, 19, ...

d = a₂ − a₁ = 9 − 4 = 5

Check: 14 − 9 = 5 ✓    19 − 14 = 5 ✓

Answer: d = 5

Example 2: Find the 10th term of 3, 7, 11, 15, ...

Given: a₁ = 3, d = 4, n = 10

a₁₀ = a₁ + (n − 1)d
a₁₀ = 3 + (10 − 1)(4)
a₁₀ = 3 + (9)(4)
a₁₀ = 3 + 36 = 39

Answer: a₁₀ = 39

Example 3: Find the 5th term of 2, 5, 8, 11, ...

Given: a₁ = 2, d = 3, n = 5

a₅ = a₁ + (n − 1)d
a₅ = 2 + (5 − 1)(3)
a₅ = 2 + (4)(3)
a₅ = 2 + 12 = 14

Answer: a₅ = 14

Example 4: Find the common difference if a₁ = 6 and a₅ = 22

Use: a_n = a₁ + (n − 1)d
22 = 6 + (5 − 1)d
22 = 6 + 4d
22 − 6 = 4d
16 = 4d
d = 4

Answer: d = 4

Guided Practice

Try It Yourself!

Answer all 3 guided questions. Hints and solutions are available if you need them!

NOTE: Correct answers will turn GREEN so you know your answer is correct!

Mastery Check

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Made for students who want to master Arithmetic Sequence