Arithmetic sequence
Identify and work with arithmetic sequences in Grade 9 algebra by finding the constant common difference between terms and using the formula aₙ=a₁+(n-1)d to find any term in the sequence.
Key Concepts
Property An arithmetic sequence is a sequence that has a constant difference between two consecutive terms, which is called the common difference.
Examples The sequence $7, 12, 17, 22, \dots$ is arithmetic because the common difference is $5$. The sequence $7, 4, 1, 2, \dots$ is arithmetic because the common difference is $ 3$. The sequence $3, 6, 12, 24, \dots$ is not arithmetic because the difference changes from $3$ to $6$.
Explanation Think of it as marching with a steady beat! Each number is a step, and the 'common difference' is the fixed distance between each step. To see if a sequence is playing this tune, just subtract any term from the one that comes right after it and check if the result is always the same.
Common Questions
What is an arithmetic sequence?
An arithmetic sequence is a list of numbers where the difference between consecutive terms is constant. This constant is called the common difference d. For 3, 7, 11, 15, the common difference is 4.
How do you find the nth term of an arithmetic sequence?
Use the formula aₙ = a₁ + (n-1)d, where a₁ is the first term, d is the common difference, and n is the term number. For the sequence 3, 7, 11, ... the 10th term is a₁₀ = 3 + (10-1)(4) = 3 + 36 = 39.
How do arithmetic sequences differ from geometric sequences?
Arithmetic sequences add the same constant (common difference) to each term. Geometric sequences multiply by the same constant (common ratio). For 2, 5, 8, 11 (arithmetic, d=3) versus 2, 6, 18, 54 (geometric, r=3).