Series I: A 21 35 X 75 101 Series II: 7 B 40 Y 369 1111 Series III: n 11 39 105 Z 453 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 99997 is divisible by 2) Find the value of [(A + B + n) + X + (Z −

ATQ, A and B are twin prime numbers and A = 6n − 1, B = 6n + 1. n is prime and n × 99997 is divisible by 2. Since 99997 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, 21, 35, X, 75, 101 Pattern: +10, +14, +18, +22, +26 (AP with common difference 4) 11 + 10 = 21 21 + 14 = 35 35 + 18 = 53 (value of X) 53 + 22 = 75 75 + 26 = 101 So, X = 53. Series II: 7, B, 40, Y, 369, 1111 We have B = 13. Pattern: 13×3 + 1 = 40 40×3 + 2 = 122 (value of Y) 122×3 + 3 = 369 369×3 + 4 = 1111 So, Y = 122. Series III: n, 11, 39, 105, Z, 453 We have n = 2. Pattern: 2 → 11: 2³ + 3 = 8 + 3 = 11 11 → 39: 11 + 3³ + 1 = 11 + 27 + 1 = 39 39 → 105: 39 + 4³ + 2 = 39 + 64 + 2 = 105 105 → Z: 105 + 5³ + 3 = 105 + 125 + 3 = 233 233 → 453: 233 + 6³ + 4 = 233 + 216 + 4 = 453 So, Z = 233. Now compute the required expression: A + B + n = 11 + 13 + 2 = 26 (A + B + n) + X = 26 + 53 = 79 Z − Y = 233 − 122 = 111 Total = 79 + 111 = 190