Geometric Sequences and Series

Develop common ratio, finite geometric sum, convergent infinite sum.

Lesson 2 of 33 parts Practice Flashcards

1/3

Common Ratio

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

A geometric sequence multiplies by a fixed r from term to term, so aₙ=a₁r^(n−1). Zero terms can make ratio comparisons undefined. Arithmetic adds a fixed amount instead. 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

$aₙ=a₁r^(n−1)$

Worked examples

Try itA geometric sequence starts at 5 with ratio 3. Find term 8.Show answer

Step 1: Term 8 has 7 multiplications by 3. Step 2: Compute 5·3^7. Step 3: The term is 10935.

Swipe sideways to move between sections

Common Ratio

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

Formulas

$aₙ=a₁r^(n−1)$

Worked examples

Try itA geometric sequence starts at 5 with ratio 3. Find term 8.Show answer

Step 1: Term 8 has 7 multiplications by 3. Step 2: Compute 5·3^7. Step 3: The term is 10935.

Finite Geometric Sum

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

For r≠1, Sₙ=a₁(1−r^n)/(1−r); if r=1, the sum is n a₁. State the number of terms. The usual formula would divide by zero. 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

$Sₙ=a₁(1−rⁿ)/(1−r), r≠1$

Worked examples

Try itFind the sum of the first 7 terms of 1+2+4+… .Show answer

Step 1: This is geometric with first term 1 and ratio 2. Step 2: Use Sₙ=(2^7−1)/(2−1). Step 3: The sum is 127.

Convergent Infinite Sum

Objective: Explain convergent infinite sum, apply the rule, and justify relevant restrictions.

For |r|<1, an infinite geometric series converges to a₁/(1−r); for |r|≥1 and nonzero a₁, it does not converge to a finite sum. The common ratio must have absolute value less than one. 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

$S∞=a₁/(1−r), |r|<1$

Worked examples

Try itFind the infinite geometric sum 7+7/3+7/9+… .Show answer

Step 1: The first term is 7 and the ratio is 1/3. Step 2: Because |1/3|<1, use S=a₁/(1−r). Step 3: The sum is 7/(1−1/3)=21/2.

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Previous lessonArithmetic Sequences and Series Next lessonBinomial Theorem and Applications