COMEDK · Maths · 5. Sequences and Series

A geometric progression consists of an even number of terms. If the sum of all the terms is five times the sum of the terms occupying the odd places, then the common ratio of the geometric progression is

A \(r=4\)
B \(r=6\)
C \(r=3\)
D \(r=2\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(r=4\)

Step-by-step Solution

Detailed explanation

Let the geometric progression have \(2n\) terms, given by \(a, ar, ar^2, ar^3, \dots, ar^{2n-1}\). The sum of all terms \(S\) is given by \(S = \dfrac{a(r^{2n} - 1)}{r - 1}\). The terms occupying the odd places are \(a, ar^2, ar^4, \dots, ar^{2n-2}\). This is a geometric…

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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From COMEDK

Match the reactions in List I with the final products formed as given in List II.

List I		List II
W	\(\text{C}_6\text{H}_5\text{CH}_3 \xrightarrow{\large{(i)\text{CrO}_3 + (\text{CH}_3\text{CO})_2\text{O};\ (ii)\text{H}_2\text{O} + heat}}\)	P	\(\text{R}_2\text{CO}\)
X	\(\text{RCOOC}_2\text{H}_5 \xrightarrow{\large{(i)\text{DIBAL-H};\ (ii)\text{H}_2\text{O}}}\)	Q	\(\text{C}_6\text{H}_5\text{COCH}_3\)
Y	\(\text{RCN} \xrightarrow{\large{(i)\text{R-Mg-X /dry ether};\ (ii)\text{H}_3\text{O}^+}}\)	R	\(\text{C}_6\text{H}_5\text{CHO}\)
Z	\(\text{C}_6\text{H}_5\text{COCl} + (\text{CH}_3)_2\text{Cd in dry ether} \rightarrow\)	S	\(\text{RCHO}\)

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