A = 2000
B = A - 999 = 1001
C = A + B - 998 = 2003
D = A + B + C -997 = 4007
E = A + B + C + D -996 = 8015
from above values we found that
T(n) = 2T(n-1) + 1 , where T(1) = 1001 and we have to find T(25)
now T(n) = 2T(n-1) + 1 ---------------------------------------(1)
T(n-1) = 2T(n-2) + 1 ----------------------------------------(2)
T(n-2) = 2T(n-3) + 1 ----------------------------------------(3)
substituting value of T(n-1) from eqn (2) in eqn(1)
T(n) = 2*2*T(n-2) + 2 + 1
substituting value of T(n-2) from eqn(3) in above eqn
T(n) = 2*2*2*T(n-3) + 4 + 2 + 1
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so on
T(n) = 2k T(n-k) + 1 + 2 + 4 + .... + 2k-1
let n-k=1 so k=n-1. Substitute this value of k in above eqn
T(n) = 2n-1 T(1) + 1 + 2 + 4 + ....... + 2n-2
T(n) = 2n-1 * 1001 + 2n-1 - 1
T(n) = 2n-1 * 1002 - 1
now substitue value of n as 25 to find Z
Z = T(25) = 224 * 1002 - 1
so value of (Z+1)/225 = (224 * 1002 - 1 + 1)/225
= 1002/2
= 501
So answer should be option 1.