Direction : Given below are two number series. Series I is a missing series…

Concept

In a consecutive-prime-square series, each term is formed by adding to the previous term the square of the next consecutive prime number. The successive differences are therefore 22, 32, 52, 72, 112, 132, … taken in strict prime order. A term is "wrong" when its difference uses a prime out of order or a non-prime.

Establish the rule from Series I

Series I (11, P, 181, 350, 639, 1000) has differences that are squares of the consecutive primes 7, 11, 13, 17, 19:

1. 181 → 350: difference 169 = 132

2. 350 → 639: difference 289 = 172

3. 639 → 1000: difference 361 = 192

Working backwards, the missing term P uses 72 then 112: P = 11 + 72 = 60, and 60 + 112 = 181. So the governing rule is “add the square of the next consecutive prime.”

Apply the rule to Series II

Starting from 242 and adding squares of consecutive primes in order 2, 3, 5, 7, 11, 13 gives the correct series:

1. 242 + 22 = 242 + 4 = 246

2. 246 + 32 = 246 + 9 = 255

3. 255 + 52 = 255 + 25 = 280

4. 280 + 72 = 280 + 49 = 329

5. 329 + 112 = 329 + 121 = 450

6. 450 + 132 = 450 + 169 = 619

Correct Series II: 242, 246, 255, 280, 329, 450, 619.

Find the wrong number and the result

The given series shows 251 in the second position, but the rule requires 242 + 22 = 246. The given series mistakenly added 32 (=9) before 22 (=4), swapping the first two prime squares. Hence the wrong number is y = 251.

Cross-check

Replace 251 with 246 and every remaining difference (9, 25, 49, 121, 169) lines up as 32, 52, 72, 112, 132 — a clean consecutive-prime-square run, confirming 251 is the sole intruder.

Therefore 2y + 1 = 2 × 251 + 1 = 503.