Algebra · StudentHub Lesson
Arithmetic Sequences
Identify arithmetic sequences and find any term using the common difference.
What you will learn
- Define an arithmetic sequence
- Identify the common difference
- Find a specific term using the explicit formula
- Distinguish arithmetic from non-arithmetic sequences
- Write a sequence given the first term and common difference
Watch the lesson
Introduction to arithmetic sequences | Sequences, series and induction · Khan Academy
Watch on YouTubeTopic notes
Main Idea
An arithmetic sequence is a list of numbers where each term is found by adding a fixed number (the common difference) to the previous term.
Key Concepts
- The common difference (d) stays the same between consecutive terms
- Sequences can be increasing (d>0) or decreasing (d<0)
- The nth term can be found directly without listing all previous terms
Definitions
- Sequence: an ordered list of numbers following a pattern
- Term: each number in a sequence
- Common difference: the fixed amount added between terms
Formulas
- nth term: aₙ = a₁ + (n-1)d
- a₁ is the first term, d is the common difference
Examples
- Sequence 2, 5, 8, 11...: d=3, a₁=2, so the 10th term = 2+(10-1)(3)=29
- Sequence 20, 15, 10...: d=-5
Common Mistakes
- Using n instead of (n-1) in the formula
- Assuming any pattern is arithmetic without checking the difference
- Sign errors with negative common differences
Key concepts
Important terms
- Sequence
- An ordered list of numbers following a rule.
- Common difference
- The constant amount added between consecutive terms.
- Term
- A single number in a sequence.
Worked examples
Problem
Find the 6th term of 4, 7, 10, 13...
- 1. Identify a₁=4, d=3
- 2. Use aₙ=a₁+(n-1)d
- 3. a₆=4+(6-1)(3)=4+15
Answer: 19
Problem
Find the common difference of 30, 25, 20, 15
- 1. Subtract consecutive terms: 25-30=-5
- 2. Confirm: 20-25=-5
Answer: d = -5
Quick revision
- The common difference is constant
- aₙ = a₁ + (n-1)d
- Check the difference between several pairs
- Sequences can decrease with negative d
- The first term is a₁, not a₀
- The formula avoids listing all terms
Check your understanding
Question 1 · Multiple choice
What is the common difference in 3, 8, 13, 18?
Question 2 · Multiple choice
Find the 5th term of the sequence with a₁=2, d=4.
Question 3 · Short answer
Find the 10th term of 1, 4, 7, 10...
Question 4 · Short answer
Is 2, 4, 8, 16 an arithmetic sequence? Why or why not?
Question 5 · True or false
An arithmetic sequence must have a constant common difference.
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