Question Statement:
Find the missing group of letters in the following series:
BC, FGH, LMNO, _____
Options:
-
(A) UVWXY
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(B) TUVWX
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(C) STUVW
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(D) RSTUV
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2018 Q.3)**
**1. Intuitive Physical Setup & Given Parameters**:
- Given series: $\text{BC}, \text{FGH}, \text{LMNO}, \dots$
- Each term consists of consecutive letters from the English alphabet.
**2. Step-by-Step Mathematical Derivation**:
1. **Term Length Progression**:
- Term 1: $\text{BC}$ (length = 2 letters)
- Term 2: $\text{FGH}$ (length = 3 letters)
- Term 3: $\text{LMNO}$ (length = 4 letters)
- Term 4: Must have length = $4 + 1 = 5$ letters.
2. **Letter Position and Gap Analysis**:
- Position of letters:
- $\text{B}=2, \text{C}=3$
- Skip: $\text{D}(4), \text{E}(5)$ (2 letters skipped)
- $\text{F}=6, \text{G}=7, \text{H}=8$
- Skip: $\text{I}(9), \text{J}(10), \text{K}(11)$ (3 letters skipped)
- $\text{L}=12, \text{M}=13, \text{N}=14, \text{O}=15$
- Skip: Next skip must be $3 + 1 = 4$ letters:
- Skip $\text{P}(16), \text{Q}(17), \text{R}(18), \text{S}(19)$
3. **Forming Next Term**:
- Starting from letter 20: $\text{T}(20)$
- Next 5 consecutive letters: $\text{T}, \text{U}, \text{V}, \text{W}, \text{X}$
- String: $\text{TUVWX}$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** $\text{UVWXY}$: ❌ Incorrect. Skips 5 letters instead of 4.
- **(B)** $\text{TUVWX}$: ✅ **CORRECT**. Skips 4 letters (P, Q, R, S) and contains 5 consecutive letters starting at T.
- **(C)** $\text{STUVW}$: ❌ Incorrect. Skips only 3 letters.
- **(D)** $\text{RSTUV}$: ❌ Incorrect. Skips only 2 letters.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: For alphanumeric series, verify both the internal length increment and the inter-group skip pattern.