COMEDK · Maths · 5. Sequences and Series

Consider an infinite geometric series with first term '\(a\)' and common ratio '\(r\)'.
If the sum of infinite geometric series is 4 and the second term is \(\dfrac{3}{4}\) then

A \(a=1 \quad r=-\dfrac{3}{4}\)
B \(a=-1 \quad r=\dfrac{3}{4}\)
C \(a=3 \quad r=\dfrac{1}{4}\)
D \(a=-3 \quad r=-\dfrac{1}{4}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(a=3 \quad r=\dfrac{1}{4}\)

Step-by-step Solution

Detailed explanation

The sum of an infinite geometric series is given by \(S = \dfrac{a}{1-r} = 4\), where \(|r| < 1\). The second term of the series is \(ar = \dfrac{3}{4}\). From the first equation, \(a = 4(1-r)\). Substituting this into the second equation: \(4(1-r)r = \dfrac{3}{4}\)…

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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Consider an infinite geometric series with first term '\(a\)' and common ratio '\(r\)'.If the sum of infinite geometric series is 4 and the second term is \(\dfrac{3}{4}\) then

Step-by-step Solution

Consider an infinite geometric series with first term '\(a\)' and common ratio '\(r\)'.
If the sum of infinite geometric series is 4 and the second term is \(\dfrac{3}{4}\) then