Direction: Given below are two number series. Series I is a missing series…
2021
Direction: Given below are two number series. Series I is a missing series while Series II is a wrong number series which follows the pattern of Series I only.
Series I: 11, P, 181, 350, 639, 1000
Series II: 242, 251, 255, 280, 329, 450, 619
If 2 is added to successive terms of a new series starting with the nearest prime of P, find the 5th term of this new series, where the candidate 5th-term values are labelled 1 = 67, 2 = 65, 3 = 69, 4 = 63.
- A.
either 1 or 3
- B.
either 2 or 3
- C.
either 2 or 4
- D.
either 3 or 4
- E.
either 1 or 2
Attempted by 3 students.
Show answer & explanation
Correct answer: A
Concept
A number series can be decoded from its consecutive differences. When those differences are themselves the squares of consecutive prime numbers, each term is the previous term plus the next prime squared. The same difference rule governs both given series, so fixing the rule on the known terms lets you recover any missing term.
Application — finding P in Series I
Series I is 11, P, 181, 350, 639, 1000. Take the differences between the known later terms, identify each as a prime squared, then walk the rule backwards to P:
181 to 350: difference 169 = 132.
350 to 639: difference 289 = 172.
639 to 1000: difference 361 = 192.
The added primes are consecutive (13, 17, 19), so the two earlier steps must use 7 and 11.
11 + 72 = 11 + 49 = 60, so P = 60. Check: 60 + 112 = 60 + 121 = 181, which matches.
Application — the new series
Next we need the nearest prime to 60. Since 60 lies exactly between 59 and 61, and both are prime at distance 1, the nearest prime is 59 or 61 — both are valid, which is why the final answer is a pair.
Form the new series by adding 2 to each successive term, starting from the nearest prime. The 5th term equals start + 2 four times = start + 8:
Starting from 59: 59, 61, 63, 65, 67. The 5th term is 67.
Starting from 61: 61, 63, 65, 67, 69. The 5th term is 69.
Result
Using the stem's sub-labelling (1 = 67, 2 = 65, 3 = 69, 4 = 63), the 5th term is 67 or 69, i.e. label 1 or label 3. So the answer is the pair “either 1 or 3”.
Cross-check
Series II (242, 251, 255, 280, 329, 450, 619) cross-checks the same prime-square mechanism. Its successive differences are 32, 22, 52, 72, 112, 132.
The genuine pattern is consecutive ODD primes squared (32, then 52, 72, 112, 132), exactly the Series I idea. The intruding 22 step is the planted error: 251 + 22 = 255 breaks the run, whereas the clean term would continue 251 + 52 onward. From 255 the tail differences 280, 329, 450, 619 do follow 52, 72, 112, 132, confirming the prime-square rule used to fix P = 60.