Given below are two number series. Series I is a missing series while series II is a wrong number series which follows pattern of series I only.

1. 11, P, 181, 350, 639, 1000
2. 242, 251, 255, 280, 329, 450, 619

If ‘y’ is the wrong number of the series II, then find the value of ‘2y+1’.

The correct option is 503.

Let us analyze the pattern of Series I to find the logic:

Series I: 11, P, 181, 350, 639, 1000

Let us calculate the differences between the consecutive known terms of Series I:
$1000-639=361={19}^2$
$639-350=289={17}^2$
$350-181=169={13}^2$

The differences are the squares of consecutive prime numbers: 13, 17, and 19. Continuing this pattern backwards, the differences before ${13}^2$ should be the squares of the prime numbers 11 and 7:
$181-P={11}^2=121\Rightarrow P=181-121=60$
$P-11=7^2=49\Rightarrow 60-11=49$ (which is correct)

Thus, the logic of the series is adding the squares of consecutive prime numbers.

Now, let us examine Series II, which follows the same logic (adding squares of consecutive prime numbers):

Series II: 242, 251, 255, 280, 329, 450, 619

Let us check the differences between the terms from right to left:
$619-450=169={13}^2$
$450-329=121={11}^2$
$329-280=49=7^2$
$280-255=25=5^2$

The consecutive prime numbers in descending order starting from 13 are: 13, 11, 7, 5, 3, and 2. Therefore, the next differences going backwards should be:
$3^2=9$
$2^2=4$

Let us apply these differences to find the correct terms starting from 255:
The term before 255 should be: $255-3^2=255-9=246$
The term before 246 should be: $246-2^2=246-4=242$ (which matches the first term)

In Series II, the term given is 251 instead of 246. Therefore, the wrong number in Series II is:
$y=251$

We need to find the value of $2y+1$:
$2y+1=2\left(251\right)+1=502+1=503$