Arithmetic Sequences and Series

Develop common difference, recursive rule, finite arithmetic sum.

Lesson 1 of 33 parts Practice Flashcards

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Common Difference

Objective: Explain common difference, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$aₙ=a₁+(n−1)d; aₙ=aₙ₋₁+d$

Worked examples

Try itAn arithmetic sequence starts at 5 with difference 3. Find term 8.Show answer

Step 1: Term 8 is 7 steps after term 1. Step 2: Compute 5+7(3). Step 3: The term is 26.

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Common Difference

Objective: Explain common difference, apply the rule, and justify relevant restrictions.

Formulas

$aₙ=a₁+(n−1)d; aₙ=aₙ₋₁+d$

Worked examples

Try itAn arithmetic sequence starts at 5 with difference 3. Find term 8.Show answer

Step 1: Term 8 is 7 steps after term 1. Step 2: Compute 5+7(3). Step 3: The term is 26.

Recursive Rule

Objective: Explain recursive rule, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$aₙ=a₁+(n−1)d; aₙ=aₙ₋₁+d$

Worked examples

Try itAn arithmetic sequence starts at 5 with difference 3. Find term 8.Show answer

Step 1: Term 8 is 7 steps after term 1. Step 2: Compute 5+7(3). Step 3: The term is 26.

Finite Arithmetic Sum

Objective: Explain finite arithmetic sum, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$Sₙ=n(a₁+aₙ)/2$

Worked examples

Try itFind the sum of 7 terms: 5, 8, ... with common difference 3.Show answer

Step 1: Use Sₙ=n[2a₁+(n−1)d]/2. Step 2: Substitute n=7, a₁=5, d=3. Step 3: The sum is 98.

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