Directions: In the following question, two quantities I and II are given. Compare quantity I and quantity II on its basis. (Only quantity is to be considered). Two series A and B are given below with one wrong term in series. A: 3, 8, 24, 45, 120, 168 B: 28, 42, 56, 98, 154, 182 Quantity I: Average of wrong terms of both the series. Quantity II: Difference between the next term of series A and the next term of series B.

### Concept & Series Pattern Analysis To evaluate sequence quantities, identify the underlying core mathematical pattern (such as square offsets or prime multipliers) driving the progression, locate the anomalous term in each series, and extrapolate the pattern to find subsequent terms. ### Step-by-Step Solution * **Analyze Series A:** $3, 8, 24, 45, 120, 168$ * Observe the relationship of terms with perfect squares: $3 = 4 - 1 = 2^2 - 1$. * $8 = 9 - 1 = 3^2 - 1$. * $24 = 25 - 1 = 5^2 - 1$. * $120 = 121 - 1 = 11^2 - 1$. * $168 = 169 - 1 = 13^2 - 1$. * The structural pattern is $p^2 - 1$ where $p$ represents consecutive prime numbers ($2, 3, 5, 7, 11, 13$). * For the missing prime $p = 7$, the term should be $7^2 - 1 = 49 - 1 = 48$. * Therefore, the wrong term in Series A is $45$ (it should be $48$). * The next prime in the sequence is $17$. The next term of Series A is $17^2 - 1 = 289 - 1 = 288$. * **Analyze Series B:** $28, 42, 56, 98, 154, 182$ * Notice these values are all multiples of $14$: $14 \times 2, 14 \times 3, 14 \times 4, 14 \times 7, 14 \times 11, 14 \times 13$. * The base multipliers ($2, 3, 7, 11, 13$) are consecutive prime numbers, except for the anomalous $4$. * The third multiplier should be the next consecutive prime number, $5$. So, the correct term is $14 \times 5 = 70$. * Therefore, the wrong term in Series B is $56$ (it should be $70$). * The next prime multiplier after $13$ is $17$. The next term of Series B is $14 \times 17 = 238$. * **Evaluate Quantities:** * Quantity I: Average of the wrong terms = $\frac{45 + 56}{2} = \frac{101}{2} = 50.5$. * Quantity II: Difference between the computed next terms = $288 - 238 = 50$. * **Comparison:** * Quantity I ($50.5$) > Quantity II ($50$). ### Exam Strategy & Shortcut Always check for prime number sequences disguised within squares, cubes, or common multipliers. They are frequently used by examiners to break standard arithmetic progression logic and test advanced pattern recognition. ### Common Pitfall Students often mistakenly assume an arithmetic difference sequence (attempting to find second or third differences) and waste valuable time when the true underlying pattern relies on prime numbers. Always test prime squares and prime multiples early. ### Final Answer Therefore, the correct answer is **Quantity I > Quantity II**.