Math Problem Statement
Use the nth term of a geometric sequence An=a1(r n-1) to answer the following questions 1.What is the 5th term of the geometric sequence 3/20,3/2,15....? 2.Find the six term of a geometric sequence where the second term is 6 and the common ratio is 2. 3.Find K so that terms k-3,k+1 and 4k-2 form a geometric sequence. 4.In the geometric sequence 6,12,24,48,...., Which term is 768? 5.The second term of a geometric is 5/4 and it's fourth term is
Solution
Let's solve each question step by step using the nth term of a geometric sequence formula:
where is the nth term, is the first term, and is the common ratio.
1. What is the 5th term of the geometric sequence 3/20, 3/2, 15...?
Step 1: Find the common ratio .
The common ratio is found by dividing the second term by the first term:
Step 2: Use the nth term formula to find the 5th term.
Here, , , and .
So, the 5th term is 1500.
2. Find the 6th term of a geometric sequence where the second term is 6 and the common ratio is 2.
Step 1: Find the first term .
Given the second term () is 6:
Step 2: Use the nth term formula to find the 6th term.
Now, , , and .
So, the 6th term is 96.
3. Find so that the terms , , and form a geometric sequence.
For these terms to form a geometric sequence, the ratio of consecutive terms must be equal:
Cross-multiply to solve:
Expand both sides:
Simplify:
Solve this quadratic equation using the quadratic formula:
So, or .
4. In the geometric sequence 6, 12, 24, 48,...., which term is 768?
Step 1: Find the common ratio .
Step 2: Use the nth term formula to find .
Solve for :
So, the 8th term is 768.
5. The second term of a geometric sequence is and the fourth term is...
Let’s denote the first term as , the common ratio as .
Given:
To proceed, please provide the fourth term to find the rest of the values.
Would you like further details or have any questions?
Here are some related questions you might consider:
- What is the 7th term of a geometric sequence given the first term and common ratio?
- How can we find the sum of the first n terms of a geometric sequence?
- How does the common ratio affect the growth of a geometric sequence?
- What happens if the common ratio is a fraction or negative?
- How to determine if a sequence is geometric given a set of terms?
Tip: In a geometric sequence, each term is obtained by multiplying the previous term by a constant (the common ratio). Understanding the common ratio helps to quickly identify the pattern of the sequence!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Geometric Sequences
Formulas
nth term of a geometric sequence
Theorems
-
Suitable Grade Level
Grades 9-12
Related Recommendation
Find the nth Term in Geometric and Arithmetic Sequences
Solving Geometric Sequence Problems with Step-by-Step Solutions
Geometric Sequences: Common Ratio and Sum Calculation
Find the nth Term and Sum of Arithmetic and Geometric Series
Solving Arithmetic and Geometric Series with nth Term and Sum Formulas