Practice Problems With Solutions On Common Ratio Types

Understanding common ratio types is essential for mastering geometric progressions and related mathematical concepts. Practice problems help reinforce these ideas and improve problem-solving skills. Below are several practice problems with detailed solutions to help students grasp various ratio types effectively.

Problem 1: Find the Common Ratio of a Geometric Sequence

Given the sequence 3, 6, 12, 24, determine the common ratio.

Solution: To find the common ratio (r), divide the second term by the first term:

r = 6 / 3 = 2

Verify with the next terms:

12 / 6 = 2, 24 / 12 = 2

Hence, the common ratio is 2.

Problem 2: Find the Next Term in a Geometric Sequence

Sequence: 5, 15, 45, 135. Find the next term.

Solution: First, find the common ratio:

r = 15 / 5 = 3

Check with other terms:

45 / 15 = 3, 135 / 45 = 3

The next term is obtained by multiplying the last term by the ratio:

135 × 3 = 405

So, the next term is 405.

Problem 3: Determine if a Sequence is Geometric

Sequence: 8, 16, 32, 64. Is this a geometric sequence? If yes, find the common ratio.

Solution: Calculate the ratio between consecutive terms:

16 / 8 = 2, 32 / 16 = 2, 64 / 32 = 2

Since the ratio is consistent, the sequence is geometric with a common ratio of 2.

Problem 4: Find the First Term Given the Common Ratio and a Later Term

In a geometric sequence, the 4th term is 48, and the common ratio is 3. Find the first term.

Solution: Use the formula for the nth term:

a_n = a₁ × r^n-1

Plug in known values:

48 = a₁ × 3^4-1 = a₁ × 3³ = a₁ × 27

Solve for a₁:

a₁ = 48 / 27 = 16 / 9

Thus, the first term is 16/9.

Problem 5: Find the Common Ratio for a Geometric Sequence

Sequence: 7, 14, 28, 56. Find the common ratio.

Solution: Divide the second term by the first:

r = 14 / 7 = 2

Check with other terms:

28 / 14 = 2, 56 / 28 = 2

The common ratio is 2.