Question

Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): The sequence \(\frac{1}{3}, -\frac{1}{2}, \frac{3}{4}, -\frac{9}{8}, \ldots\) is an A.P.
Reason (R): The constant sequence is the only sequence which is both an A.P. as well as G.P. In the light of the above statements, choose the most appropriate answer from the options given below:

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

Always check differences for A.P. and ratios for G.P. If neither is constant, it is neither A.P. nor G.P.

Answer 1

The Correct Option is D

Concept:
• Arithmetic Progression (A.P.): Difference between consecutive terms is constant.
• Geometric Progression (G.P.): Ratio between consecutive terms is constant.

Step 1: Check Assertion (A). Given sequence: \[ \frac{1}{3},\ -\frac{1}{2},\ \frac{3}{4},\ -\frac{9}{8}, \ldots \] Find differences: \[ -\frac{1}{2} - \frac{1}{3} = -\frac{3+2}{6} = -\frac{5}{6} \] \[ \frac{3}{4} - \left(-\frac{1}{2}\right) = \frac{3}{4} + \frac{1}{2} = \frac{3+2}{4} = \frac{5}{4} \] Differences are not equal. So, it is not an A.P. Thus Assertion (A) is false.

Step 2: Check Reason (R). A sequence which is both A.P. and G.P.:
• In A.P.: common difference \(d\)
• In G.P.: common ratio \(r\) For both to hold: \[ a,\ a,\ a,\ a, \ldots \] i.e., constant sequence. Thus Reason (R) is true.

Step 3: Conclusion. \[ \boxed{(4)\ (A)\ \text{is not correct but}\ (R)\ \text{is correct}} \]