Series I: A 18 29 X 63 86 Series II: 10 B 53 Y 859 3440 Series III: n 11 27 61 Z 273 Note: A and B are twin prime numbers. A = 6n − 1 B = 6n + 1 (n is also a prime number and the multiple of n and 88889 is divisible by 2) Find the value of [X + Y + Z − (A + B +
n)].

ATQ, A and B are twin prime numbers and A = 6n − 1, B = 6n + 1. n is prime and n × 88889 is divisible by 2. Since 88889 is odd, for their product to be even, n must be even. The only even prime is 2. So, n = 2. Then A = 11 and B = 13. Series I: A, 18, 29, X, 63, 86 Pattern: +7, +11, +15, +19, +23 (odd numbers increasing by 4) 11 + 7 = 18 18 + 11 = 29 29 + 15 = 44 (value of X) 44 + 19 = 63 63 + 23 = 86 So, X = 44. Series II: 10, B, 53, Y, 859, 3440 We have B = 13. Pattern: 13×4 + 1 = 53 53×4 + 2 = 214 (value of Y) 214×4 + 3 = 859 859×4 + 4 = 3440 So, Y = 214. Series III: n, 11, 27, 61, Z, 273 We have n = 2. Pattern: 2 → 11: 2×4 + 3 = 11 11 → 27: 11×2 + 5 = 27 27 → 61: 27×2 + 7 = 61 61 → Z: 61×2 + 9 = 131 131 → 273: 131×2 + 11 = 273 So, Z = 131. Now compute the required expression: X + Y + Z = 44 + 214 + 131 = 389 A + B + n = 11 + 13 + 2 = 26 X + Y + Z − (A + B + n) = 389 − 26 = 363