Concept 2 of 19Foundation
Watch on YouTubeVideoArithmetic sequences (add the same)
An arithmetic sequence adds or subtracts the same number every step. That fixed number is the common difference (d).
- 3, 6, 9, 12, 15, 18… (d = +3)
- 20, 17, 14, 11, 8… (d = −3)
- 7, 12, 17, 22, 27… (d = +5)
nth-term formula: aₙ = a + (n − 1) × d
Example
20th term of 5, 9, 13, 17, … (d = 4):
a₂₀ = 5 + (20 − 1)·4 = 5 + 76 = 81.
a₂₀ = 5 + (20 − 1)·4 = 5 + 76 = 81.
💡 Tip:Same first-difference every time? It's arithmetic.
▸Why does this work? (derivation)
Why aₙ = a + (n−1)d? Start at a. After 1 step you've added d ONCE. After 2 steps, added d twice. After (n−1) steps, added d (n−1) times. So aₙ = a + (n−1)·d.
🎯 Try it!
5 questions to check what you just read.
0 / 5
- Q1.Common difference of 11, 17, 23, 29?
- Q2.10th term of 2, 5, 8, 11, …?
- Q3.Next term of 45, 40, 35, 30, …?
- Q4.20th term of 1, 4, 7, 10, …?
- Q5.100th term of 10, 12, 14, 16, …?