COMEDK · Maths · 5. Sequences and Series

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 \(\dfrac{32}{81}\), then which of the following is not true

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

Verified Solution

Answer & Solution

Correct Answer

(B) \(S_{\infty}=3+2 \sqrt{2}\)

Step-by-step Solution

Detailed explanation

Let the first term be \(a = 4\) and the common ratio be \(r\), where \(0 < r < 1\). The terms of the geometric progression are \(a, ar, ar^2, ar^3, ar^4, \dots\). The third term is \(T_3 = ar^2 = 4r^2\) and the fifth term is \(T_5 = ar^4 = 4r^4\). Given…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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