Subtopic: Fundamentals of Biomolecules, enzyme kinetics
Question Statement:
Which of the following are geometric series?
P. 1, 6, 11, 16, 21, 26, ...
Q. 9, 6, 3, 0, -3, -6, ...
R. 1, 3, 9, 27, 81, ...
S. 4, -8, 16, -32, 64, ...
Options:
(A) P and Q only
(B) R and S only
(C) Q and S only
(D) P, Q and R only
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2019 Q.19)**
**1. Governing Concept**:
- **Topic**: Quantitative Aptitude & Geometric Progressions (GP)
- A sequence $a_1, a_2, a_3, \dots$ is a **geometric series/progression** if the ratio of consecutive terms is a constant non-zero common ratio $r = \frac{a_{n+1}}{a_n}$:
- Sequence P: $1, 6, 11, 16, 21, \dots \implies$ difference is constant ($d = +5$). This is an **Arithmetic Progression (AP)**.
- Sequence Q: $9, 6, 3, 0, -3, \dots \implies$ difference is constant ($d = -3$). This is an **Arithmetic Progression (AP)**.
- Sequence R: $1, 3, 9, 27, 81, \dots \implies \frac{3}{1} = \frac{9}{3} = \frac{27}{9} = 3$. Constant common ratio $r = 3 \implies$ **Geometric Progression (GP)**.
- Sequence S: $4, -8, 16, -32, 64, \dots \implies \frac{-8}{4} = \frac{16}{-8} = -2$. Constant common ratio $r = -2 \implies$ **Geometric Progression (GP)**.
- Therefore, **R and S only** are geometric series.
**2. Option Analysis**:
- **Option (A) is INCORRECT**: P and Q are arithmetic, not geometric.
- **Option (B) [R and S only] is CORRECT**: Both sequences exhibit invariant geometric multipliers ($r = 3$ and $r = -2$).
- **Option (C) is INCORRECT**: Q is an AP.
- **Option (D) is INCORRECT**: P and Q are APs.
**3. Final Answer**:
- **Correct Choice**: **B**
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