Two series I and II given below each have one wrong number P and Q respectively. Series I: 5, 20, 40, 67, 105, 140, 224 Series II: 13, 20, 30, 98, 119, 334 Find the difference between P and Q

### Concept & Pattern Recognition To find a wrong number in a series, we calculate the differences between consecutive terms. If the first differences don't show a clear pattern, we calculate the "second differences" (the differences of the differences). Often, these reveal standard sequences like prime numbers, squares, or cubes. ### Step-by-Step Solution **Step 1: Analyze Series I to find P** Given Series I: $5, 20, 40, 67, 105, 140, 224$ Let's calculate the first level of differences between consecutive terms: * $20 - 5 = 15$ * $40 - 20 = 20$ * $67 - 40 = 27$ * $105 - 67 = 38$ * $140 - 105 = 35$ * $224 - 140 = 84$ First differences: $15, 20, 27, 38, 35, 84$. Now, let's calculate the second level of differences (for the first few terms): * $20 - 15 = 5$ * $27 - 20 = 7$ * $38 - 27 = 11$ Notice the sequence $5, 7, 11$. These are consecutive prime numbers! Following this pattern, the next prime numbers should be $13$ and $17$. Let's reconstruct the first differences using these next primes: * Next first difference should be: $38 + 13 = 51$ * The first difference after that should be: $51 + 17 = 68$ Now, let's apply these corrected first differences to find the correct terms in the main series: * Correct 6th term: $105 + 51 = 156$ * Let's verify the final term: $156 + 68 = 224$. This matches the last term of the series perfectly! Therefore, the number $140$ is incorrect and should be $156$. So, **$P = 140$**. **Step 2: Deduce Q using the given options** We need to find the difference $P - Q$. Let's test the given options against $P = 140$ to see which one yields a number that actually exists in Series II ($13, 20, 30, 98, 119, 334$). * If $P - Q = 24$, then $140 - Q = 24 \implies Q = 116$ (Not in Series II) * If $P - Q = 36$, then $140 - Q = 36 \implies Q = 104$ (Not in Series II) * If $P - Q = 38$, then $140 - Q = 38 \implies Q = 102$ (Not in Series II) * If $P - Q = 42$, then $140 - Q = 42 \implies Q = 98$ (This number **is** in Series II!) Since $98$ is a term in Series II, it must be the wrong number $Q$. ### Exam Strategy & Shortcut **Reverse Engineering:** In competitive exams, time is your most valuable asset. Once you confidently find $P = 140$, do not waste time figuring out the complex pattern of Series II. The wrong number $Q$ *must* be one of the numbers explicitly listed in Series II. By subtracting the multiple-choice options from $140$, you instantly find that only option (d) results in a number ($98$) that is present in the second series. ### Common Pitfall A common mistake is stubbornly trying to decode the exact mathematical pattern of the second series when it's not strictly necessary to answer the multiple-choice question. Always let the options guide your logic when you hit a roadblock. ### Final Answer Therefore, the correct answer is **42**.