Question

The terms of an infinitely decreasing geometric progression in which all the terms are positive, the first term is 4, and the difference between third and fifth term is \( \frac{32}{81} \), then which of the following is not true?

• A. \( S_{\infty} = 3 + 2\sqrt{2} \)
• B. \( r = \frac{1}{3} \)
• C. \( S_{\infty} = 6 \)
• D. \( r = \frac{2\sqrt{2}}{3} \)

Hint:

In a geometric progression, the sum of the infinite series is found using \( S_{\infty} = \frac{a}{1 - r} \) when \( |r| < 1 \).

Answer 1

The Correct Option is A

Step 1: General form of geometric progression.
In a geometric progression, the \( n \)-th term is given by:
\[ T_n = a r^{n-1} \] where \( a \) is the first term and \( r \) is the common ratio.

Step 2: Using the given information.
We are given that the first term \( a = 4 \) and the difference between the third and fifth terms is \( \frac{32}{81} \).
Thus, the third term is \( T_3 = 4r^2 \), and the fifth term is \( T_5 = 4r^4 \). The difference is:
\[ T_5 - T_3 = 4r^4 - 4r^2 = \frac{32}{81} \] Factor the left-hand side:
\[ 4r^2(r^2 - 1) = \frac{32}{81} \] Simplifying:
\[ r^2(r^2 - 1) = \frac{8}{81} \]

Step 3: Solving the quadratic equation.
We now solve the equation \( r^2(r^2 - 1) = \frac{8}{81} \). Solving for \( r \), we find:
\[ r = \frac{1}{3} \]

Step 4: Finding the sum of the infinite geometric series.
The sum of an infinite geometric progression is given by:
\[ S_{\infty} = \frac{a}{1 - r} \] Substituting the values \( a = 4 \) and \( r = \frac{1}{3} \):
\[ S_{\infty} = \frac{4}{1 - \frac{1}{3}} = \frac{4}{\frac{2}{3}} = 6 \]

Step 5: Checking the options.
- (A) \( S_{\infty} = 3 + 2\sqrt{2} \): This is incorrect because we found \( S_{\infty} = 6 \).
- (B) \( r = \frac{1}{3} \): This is correct.
- (C) \( S_{\infty} = 6 \): This is correct.
- (D) \( r = \frac{2\sqrt{2}}{3} \): This is incorrect based on the calculated value of \( r \).

Step 6: Final Answer.
The correct answer is option (A), as the sum \( S_{\infty} = 3 + 2\sqrt{2} \) is not true.