Given below three series I, II & III given and each series has a wrong number. The number that should come in place of the wrong number in series I, II & III is P, Q & R respectively. Find the value of P, Q & R and then find which of the following statement is true. 244, 380, 476, 536, 568, 580, 580 160, 200, 480, 1200, 3600, 12600, 50400 15, 20, 36, 65, 111, 186, 264

### Concept & Logical Deduction To find the wrong numbers in these series, we must identify the underlying mathematical pattern (differences, double differences, or multipliers) for each sequence and replace the anomaly to satisfy the pattern. ### Step-by-Step Solution * **Series I: 244, 380, 476, 536, 568, 580, 580** * Calculate the differences between consecutive terms from right to left: $580 - 580 = 0$, $580 - 568 = 12$, $568 - 536 = 32$, $536 - 476 = 60$, $476 - 380 = 96$. * Calculate the double differences: $12 - 0 = 12$, $32 - 12 = 20$, $60 - 32 = 28$, $96 - 60 = 36$. * The double differences ($12, 20, 28, 36$) form an arithmetic progression with a common difference of $8$. * The next double difference should be $36 + 8 = 44$. * The next single difference should be $96 + 44 = 140$. * The first term should be $380 - 140 = 240$, not $244$. * Thus, $P = 240$. * **Series II: 160, 200, 480, 1200, 3600, 12600, 50400** * Calculate the ratio of consecutive terms: $50400 / 12600 = 4$, $12600 / 3600 = 3.5$, $3600 / 1200 = 3$, $1200 / 480 = 2.5$. * The multipliers follow the pattern: $1.5, 2, 2.5, 3, 3.5, 4$. * Let's test this: $160 \times 1.5 = 240$. * $240 \times 2 = 480$ (This matches the third term). * Therefore, $200$ is the wrong term and should be $240$. * Thus, $Q = 240$. * **Series III: 15, 20, 36, 65, 111, 186, 264** * Calculate the differences: $20 - 15 = 5$, $36 - 20 = 16$, $65 - 36 = 29$, $111 - 65 = 46$. * Calculate the double differences: $16 - 5 = 11$, $29 - 16 = 13$, $46 - 29 = 17$. * The double differences are consecutive prime numbers: $11, 13, 17$. * The next prime numbers are $19$ and $23$. * The next differences should be $46 + 19 = 65$, and $65 + 23 = 88$. * The next terms should be $111 + 65 = 176$, and $176 + 88 = 264$. * Therefore, $186$ is the wrong term and should be $176$. * Thus, $R = 176$. * **Conclusion:** We have $P = 240$, $Q = 240$, and $R = 176$. This gives the relation $P = Q > R$. ### Exam Strategy & Shortcut Instead of solving from left to right, start from the right end of a sequence if the numbers are easier to manage or if the pattern becomes clearer. In Series I and II, working backwards quickly revealed the arithmetic and geometric progressions. ### Common Pitfall A common mistake is trying to fit a complex polynomial or power series when a simple double difference or multiplier pattern exists. Always check first and second differences first. ### Final Answer Therefore, the correct answer is **P = Q > R**.