f(n)=2^n- 1
For example:
When we find out how many times the N plates of the Tower of Hanoi must move, we get the following recursive formula:
a[ 1]= 1;
a[n]= a[n- 1]* 2+ 1;
Reference general formula
a[ 1]= 1;
a[n]= a[n- 1]* 2+ 1;
You can get a [I] = 2i-1;
Proved by mathematical induction:
Guess a [I] = 2 I- 1
When i= 1, it is obviously true.
If i=k holds, that is, a [k] = 2k-1; Then:
From a [n] = a [n-1] * 2-1; get
a[k+ 1] = a[k] * 2 - 1
= 2^k * 2 - 1
= 2^(k- 1) - 1
Therefore, it is proved that.
Extended data:
N-order Vantage moves: Let the number of gold pieces be n, then the number of moves =2 n-1 power.
Problem background: There are three pillars A, B and C. At first, N disks of different sizes were stacked on pillar A from small to large. Now, all disks should be moved from A to C, and the small disk should always be above the big disk during the movement. Find out the least number of times to move the plate.
(1) Move disk A and disk B to the two-disk puzzle in the second column;
(2) The single disk problem of moving disk C to 3 columns;
(3) The double disk problem of moving disk A and disk B to three columns.
As can be seen from the above, every simplified problem is easier than the original problem, so the problem will become an easy-to-solve original problem.
Baidu encyclopedia-Fanta problem
Solution: as shown on the right, connect EF, FG, GH, EH,
E and h are the midpoint of AB and DA, respectively,
Eh is the cent