Math Problem Statement
Given a geometric sequence whose 1st term = 48 and 4th term = 384.
Find the 7th term.
Solution
We are given a geometric sequence with the following information:
- The 1st term () = 48
- The 4th term () = 384
To solve for the 7th term, we will use the formula for the -th term of a geometric sequence:
Where:
- is the -th term
- is the first term
- is the common ratio
- is the term number
Step 1: Use the formula to express the 4th term
For the 4th term:
Substitute the known values:
Step 2: Solve for
Divide both sides by 48:
Now take the cube root of both sides:
Step 3: Find the 7th term
Now that we know , we can find the 7th term () using the formula:
Calculate :
Thus, the 7th term is 3072.
Would you like more details on any part of the solution, or do you have other questions?
Here are some related questions to explore:
- How do you derive the common ratio from a geometric sequence if you only know two terms?
- What happens if the common ratio is negative or a fraction?
- How would the formula change if we were looking for the 10th term instead?
- Can you prove why the formula for the -th term of a geometric sequence works?
- How does the sum of the first terms of a geometric sequence differ from the -th term?
Tip: The common ratio in a geometric sequence determines the growth or decay rate, and if it's greater than 1, the terms increase; if less than 1, the terms decrease.
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Math Problem Analysis
Mathematical Concepts
Geometric Sequences
Exponents
Algebra
Formulas
a_n = a_1 * r^(n-1)
Theorems
Geometric Sequence Formula
Suitable Grade Level
Grades 7-10
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