The value of Z+1/225 = _

Question

A = 2000
B = A – 999
C = A + B – 998
D = A + B + C – 997
.
.
.
 Z= A + B + C … + X – 975
The value of Z+1/2²⁵ = _______

1.   501
2.   901
3.   801
4.   2048

Comments

i was trying to qrite the equation in terms of addition of A and subtracting (999 + 998 + 997 + ... 975) i.e 24675

but not able to write in terms of A as there is no pattern.

Answer 1

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

.

.

.

so on

T(n) = 2^k T(n-k) + 1 + 2 + 4 + .... + 2^k-1

let n-k=1 so k=n-1. Substitute this value of k in above eqn

T(n) = 2^n-1 T(1) + 1 + 2 + 4 + ....... + 2^n-2

T(n) = 2^n-1 * 1001 + 2^n-1 - 1

T(n) = 2^n-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. 

Comments

yes..answer is 501

thanks a lot