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 3227\frac{32}{27} 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:

  1. Set up the sequence:

    Let the geometric sequence be:

    6,a1,a2,a3,32276, \, a_1, \, a_2, \, a_3, \, \frac{32}{27}

    Here, a1a_1, a2a_2, and a3a_3 are the three numbers we need to find. The first term is 6, and the last term is 3227\frac{32}{27}.

  2. Use the formula for a geometric sequence:

    In a geometric sequence, the ratio between consecutive terms is constant. Let the common ratio be rr. Then, the sequence follows this pattern:

    a1=6r,a2=6r2,a3=6r3,3227=6r4a_1 = 6r, \quad a_2 = 6r^2, \quad a_3 = 6r^3, \quad \frac{32}{27} = 6r^4

  3. Find the common ratio rr:

    The last term of the sequence gives the equation:

    6r4=32276r^4 = \frac{32}{27}

    Solve for rr:

    r4=3227÷6=3227×16=32162=1681r^4 = \frac{32}{27} \div 6 = \frac{32}{27} \times \frac{1}{6} = \frac{32}{162} = \frac{16}{81}

    Taking the fourth root of both sides:

    [ r = \sqrt[4]{\frac{16}{81}} = \frac{\sqrt[4]{16}}{\sqrt[4]{81}} = \frac{2}{3} ]

  4. Find the three numbers:

    Now that we have the common ratio r=23r = \frac{2}{3}, we can find the three geometric means:

    a1=6r=6×23=4a_1 = 6r = 6 \times \frac{2}{3} = 4 a2=6r2=6×(23)2=6×49=249=83a_2 = 6r^2 = 6 \times \left( \frac{2}{3} \right)^2 = 6 \times \frac{4}{9} = \frac{24}{9} = \frac{8}{3} a3=6r3=6×(23)3=6×827=4827=169a_3 = 6r^3 = 6 \times \left( \frac{2}{3} \right)^3 = 6 \times \frac{8}{27} = \frac{48}{27} = \frac{16}{9}

  5. Final sequence:

    The full geometric sequence is:

    6,4,83,169,32276, \, 4, \, \frac{8}{3}, \, \frac{16}{9}, \, \frac{32}{27}

Thus, the three geometric means between 6 and 3227\frac{32}{27} are 44, 83\frac{8}{3}, and 169\frac{16}{9}.


Would you like more details on any specific step or clarification?

Relative Questions:

  1. How do you derive the formula for the common ratio in a geometric sequence?
  2. What is the difference between arithmetic and geometric sequences?
  3. How can you generalize the insertion of geometric means for any two numbers?
  4. How do you calculate roots of fractions such as the fourth root of 1681\frac{16}{81}?
  5. 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