Arithmetic Sequences and Series

Explore common difference, recursive rule, finite arithmetic sum.

Lesson 1 of 53 parts Practice Flashcards

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Common Difference

Objective: Explain common difference and use it to solve or interpret a problem.

An arithmetic sequence adds a constant d at each step; the explicit rule is aₙ=a₁+(n−1)d for positive integer n. The first term is zero steps after itself.

Worked examples

Try itArithmetic Sequences and Series — practice 1.1: An arithmetic sequence starts at 5 with difference 4. Find term 6.Show answer

Step 1: Term 6 is 5 steps after term 1. Step 2: Compute 5+5(4). Step 3: The term is 25.

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Common Difference

Objective: Explain common difference and use it to solve or interpret a problem.

An arithmetic sequence adds a constant d at each step; the explicit rule is aₙ=a₁+(n−1)d for positive integer n. The first term is zero steps after itself.

Worked examples

Try itArithmetic Sequences and Series — practice 1.1: An arithmetic sequence starts at 5 with difference 4. Find term 6.Show answer

Step 1: Term 6 is 5 steps after term 1. Step 2: Compute 5+5(4). Step 3: The term is 25.

Recursive Rule

Objective: Explain recursive rule and use it to solve or interpret a problem.

A recursive rule gives an initial term and aₙ=aₙ₋₁+d for later terms. It describes how to build the list one step at a time. An initial term.

Worked examples

Try itArithmetic Sequences and Series — practice 2.1: An arithmetic sequence starts at 6 with difference 2. Find term 9.Show answer

Step 1: Term 9 is 8 steps after term 1. Step 2: Compute 6+8(2). Step 3: The term is 22.

Finite Arithmetic Sum

Objective: Explain finite arithmetic sum and use it to solve or interpret a problem.

An arithmetic series adds a finite sequence; Sₙ=n(a₁+aₙ)/2 pairs first and last terms into equal sums. Every symmetric pair has the same sum.

Worked examples

Try itArithmetic Sequences and Series — practice 3.1: Find the sum of 11 terms: 7, 10, ... with common difference 3.Show answer

Step 1: Use Sₙ=n[2a₁+(n−1)d]/2. Step 2: Substitute n=11, a₁=7, d=3. Step 3: The sum is 242.

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