Math Problem Statement
Insert three numbers (called geometric means) between 6 and 32/27, so that five numbers form a geometric sequence. and explain the process
Solution
To insert three numbers between 6 and such that the five numbers form a geometric sequence, we need to follow the process of finding the common ratio of the sequence and the intermediate terms.
Steps:
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Set up the sequence:
Let the geometric sequence be:
Here, , , and are the three numbers we need to find. The first term is 6, and the last term is .
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Use the formula for a geometric sequence:
In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be . Then, the sequence follows this pattern:
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Find the common ratio :
The last term of the sequence gives the equation:
Solve for :
Taking the fourth root of both sides:
[ r = \sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}} = \frac{2}{3} ]
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Find the three numbers:
Now that we have the common ratio , we can find the three geometric means:
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Final sequence:
The full geometric sequence is:
Thus, the three geometric means between 6 and are , , and .
Would you like more details on any specific step or clarification?
Relative Questions:
- How do you derive the formula for the common ratio in a geometric sequence?
- What is the difference between arithmetic and geometric sequences?
- How can you generalize the insertion of geometric means for any two numbers?
- How do you calculate roots of fractions such as the fourth root of ?
- What are some real-world applications of geometric sequences?
Tip:
In a geometric sequence, each term is a constant multiple of the previous term, unlike in arithmetic sequences where the difference between terms is constant.
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Math Problem Analysis
Mathematical Concepts
Geometric Sequences
Ratios
Roots of Fractions
Formulas
General formula for a geometric sequence: a_n = a_1 * r^(n-1)
Equation for the common ratio: r^4 = (last_term / first_term)
Fourth root calculation for fractions: r = sqrt[4](16/81)
Theorems
Properties of Geometric Sequences
Roots of Fractions
Suitable Grade Level
Grades 9-11