Transcript Recurrence Relation for the Towers of Hanoi

Recurrence Relation for the
Towers of Hanoi
Given: T(1) = 1
T(n) = 2 T( n-1 ) +1
N
No.Moves
1
1
2
3
3
7
4
15
5
31
T(n) = 2 T( n-1 ) +1
T(n) = 2
+1
T(n) = 2 [ 2 T(n-2) + 1 ] +1
T(n) = 2 [ 2
+ 1 ] +1
T(n) = 2 [ 2 [ 2 T(n-3) + 1 ]+ 1 ] +1
T(n) = 2 [ 2 [ 2 [ 2 T( n-4 ) + 1 ] + 1 ]+ 1 ] +1
T(n) = 24 T ( n-4 ) + 15
. . .
T(n) = 2k T ( n-k ) + 2k - 1
Since n is finite, k -> n. Therefore,
lim T(n) k -> n = 2n - 1