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:
an = a1 × rn-1
Plug in known values:
48 = a1 × 34-1 = 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.