Answered

The $3^{\text{rd}}$ and $6^{\text{th}}$ terms of a Geometric Progression (G.P.) are 48 and $14 \frac{2}{9}$, respectively. Find the $5^{\text{th}}$ term. Write down the first four terms of the G.P.

Verified Answer :

  • Identify the G.P. formula and use the given third and sixth terms to establish the relationship \(r^3 = \frac{128}{432} = \frac{8}{27}\).
  • Solve for the common ratio \(r = \frac{2}{3}\).
  • Determine the first term \(a = 108\) using the third term equation.
  • Calculate the first four terms (\(108, 72, 48, 32\)) and find the fifth term (\(21 \frac{1}{3}\)).
  • Final answer for the fifth term is \(\boxed{21 \frac{1}{3}}\).

Explanation

  1. Analysis of the Geometric Progression

A geometric progression (G.P.) is defined by a first term \(a\) and a common ratio \(r\), where the \(n\)-th term is given by \(a_n = a \cdot r^{n-1}\). We are given the third term \(a_3 = 48\) and the sixth term \(a_6 = 14 \frac{2}{9}\). Converting the mixed fraction, we get \(a_6 = \frac{14 \times 9 + 2}{9} = \frac{128}{9}\). Our goal is to find the common ratio \(r\), the first term \(a\), the fifth term \(a_5\), and list the first four terms of the sequence.

  1. Finding the Common Ratio \(r\)

Using the formula for the \(n\)-th term:

  1. The third term is \(a_3 = a \cdot r^2 = 48\).
  2. The sixth term is \(a_6 = a \cdot r^5 = \frac{128}{9}\).

Dividing the equation for the sixth term by the third term gives:
\[\frac{a \cdot r^5}{a \cdot r^2} = \frac{128/9}{48}\]
\[r^3 = \frac{128}{9 \times 48} = \frac{128}{432} = \frac{8}{27}\]
Taking the cube root of both sides, we find the common ratio:
\[r = \sqrt[3]{\frac{8}{27}} = \frac{2}{3}\]

  1. Finding the First Term and First Four Terms

Now that we have the common ratio \(r = \frac{2}{3}\), we substitute it back into the equation for the third term to find the first term \(a\):
\[a \cdot \left(\frac{2}{3}\right)^2 = 48\]
\[a \cdot \frac{4}{9} = 48\]
\[a = 48 \times \frac{9}{4} = 12 \times 9 = 108\]
The first term \(a_1\) is \(108\). We can now determine the first four terms of the G.P.:

  • \(a_1 = 108\)
  • \(a_2 = 108 \cdot \frac{2}{3} = 72\)
  • \(a_3 = 72 \cdot \frac{2}{3} = 48\)
  • \(a_4 = 48 \cdot \frac{2}{3} = 32\)
  1. Calculating the Fifth Term and Final Conclusion

Finally, we calculate the fifth term \(a_5\) by multiplying the fourth term by the common ratio \(r\):
\[a_5 = a_4 \cdot r = 32 \cdot \frac{2}{3} = \frac{64}{3} = 21 \frac{1}{3}\]
Thus, the fifth term is \(21 \frac{1}{3}\) and the first four terms are \(108, 72, 48, 32\). The final answer is \(\boxed{21 \frac{1}{3}}\).

Examples

Geometric progressions appear frequently in finance and population growth. For example, if a company's investment grows at a steady percentage rate each year, the value of the investment at the end of each year follows a geometric progression. By knowing the value at two different points in time, you can determine the annual growth rate (the common ratio) and predict the future value of the investment, similar to how we solved for the missing terms of the sequence here.

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