Geometric Sequences and Series

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

Lesson 2 of 53 parts Practice Flashcards

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Common Ratio

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

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.

Worked examples

Try itGeometric Sequences and Series — practice 1.1: A geometric sequence starts at 6 with ratio 2. Find term 6.Show answer

Step 1: Term 6 has 5 multiplications by 2. Step 2: Compute 6·2^5. Step 3: The term is 192.

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Common Ratio

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

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.

Worked examples

Try itGeometric Sequences and Series — practice 1.1: A geometric sequence starts at 6 with ratio 2. Find term 6.Show answer

Step 1: Term 6 has 5 multiplications by 2. Step 2: Compute 6·2^5. Step 3: The term is 192.

Finite Geometric Sum

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

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.

Worked examples

Try itGeometric Sequences and Series — practice 2.1: Find the sum of the first 8 terms of 1+2+4+… .Show answer

Step 1: This is geometric with first term 1 and ratio 2. Step 2: Use Sₙ=(2^8−1)/(2−1). Step 3: The sum is 255.

Convergent Infinite Sum

Objective: Explain convergent infinite sum and use it to solve or interpret a problem.

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.

Worked examples

Try itGeometric Sequences and Series — practice 3.1: Find the infinite sum 3+3/2+3/4+… .Show answer

Step 1: The first term is b and common ratio is 1/2. Step 2: Since |1/2|<1, the series converges. Step 3: Compute 3/(1−1/2)=6.

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