Example
For example in 4 discs:
Pole A Pole B Pole C
4321
432 1
43 2 1
43 21
4 21 3
41 2 3
- 32
- 321
Begins to change as the additional disc is used and an extra tower has to be rebuilt and un-built.
- 321
- 32
2 41 3
21 4 3
- 43
2 43 1
- 1
4321
This pattern occurs because in order for a 4 disc tower to be built a 3-disc tower must be built then un-built. The added 1
The 1st disc moves to pole C and creates a tower the same to the 3-disc tower. When the tower is complete, extra disc (disc 4) is moved on to pole B. This where the plus one comes from, then the tower is rebuilt on pole B.
Due to the unbuilding and rebuilding, each tower finishes on the opposite pole to the previous tower.
This is a table of the moves made by each disc.
Each disc’s number of moves doubles every time a new disc is added (as it goes down the table). The number of moves made by each disc is halved as the disc number goes up.
For example in a 3 disc tower the number of moves made by the 1st disc (4) is halved to get the number of moves made by the 2nd disc (2). This pattern occurs throughout Hanoi.
The General rule is 2n-1. This rule means 2 is to the power of the number of discs, subtract 1.
For example for a 4 disc tower:
2x2x2x2 =16
16 – 1 = 15
The difference between the number of moves doubles as the disc number occurs in descending order. Therefore to get from one number to another, 2 must multiply the initial number. As the difference requires a multiplication in sum, this means this is a geometric sequence. A geometric sequence must have an addition or subtraction present.
In the table 2) the difference between each disc’s number of moves in each tower can be expressed using nth terms.
For example in a tower of 6 discs as 16 is the most amount of moves made by a single disc.
n=16
These are numbers in a geometric sequence. Each number is to the power of 2 of the previous.
1 + 2 + 4 + 8
This shows the sequences in the nth term, where the last number is n.
n-3 n-2 n-1 n
If n is equal to 8, than the number previous in that series will be n-1. Therefore any number straight after would be n+1. For example in the n=8 sequences 16 = n+1.
To find the sum of nth terms in a geometric sequence it is possible to multiply each number against its opposite number in the sequence. All the numbers will equal the same, if that number is multiplied by the nth term and divided by 2, the sum of the numbers in the sequence will have been produced.
For example using a geometric sequence for 32 where each number is doubled to get the next number. When the numbers are multiplied by the opposite number in that sequence, the number will always be 32.
To find the formula of the general geometric sequences, the sum of the sequences is subtracted by the sum of the sequence number multiplied by the common ratio.
- Explanation in algebra of the general geometric sequences:
S = a + ar +ar2
This is the rule for each number: arn-1
This sequence is then multiplied by the common ratio ( r )
rS = ar + ar2 + ar3
This the rule: arn
Then 1. subtracted away from 2.
This gives:
(r-1)S= -a +arn
= a(rn-1)
S=a(rn-1)
(r-1)
When the n=16 sequence, the number are doubled and divided. For example:
‘The terms of a geometric sequence are formed by multiplying one term by a fixed number, the common ratio to obtain the next.’
If a1 is the first term
k is the general term
r is the common ratio
n is the number of terms
ak+1 =rak
This means the kth term is equal to the k term when it is multiplied by the common ratio.
Example
When ak is 4.
ak+1
= 4+1
= 8
r = 2
2*4 = 8
4+1 = 8 = 2*4
The Sum of the terms of a geometric sequence
I will use the 5-disc tower sequence to investigate the Sum of the terms of a geometric sequence.
1+2+4+8+16+32 = 63
1+2+4+8+16+32 = 64 –1.
The geometric series with 6 terms and common ratio 2:
1+2+4+8+16+32 = 63
1+2+4+8+16+32 = Series(s)
Then I multiplied it by the common ratio 2:
2+4+8+16+32+64 = 2S
2S = + 2 + 4 + 8 + 16 +.32 + 64
S = 1 + 2 + 4 + 8 + 16 + 32
S = -1 + 0 + 0 + 0 + 0 + 0 + 64
S = 64-1
S = 63
2S = 21+ 22 + 23 + 24 + 25 + 26
S = 1 + 2 + 4 + 8 + 16 + 32
S = -1 + 0 + 0 + 0 + 0 + 0 +26
S = 26-1
The equation is
S = rn-a1
The series is equal to the common ratio to the power of the number of terms minus the first term.
This method can obtain a formula:
Explanation in algebra of the general geometric sequences:
S = a + ar +ar2
This is the rule for each number: arn-1
This is sequence is then multiplied by the common ratio ( r )
rS = ar + ar2 + ar3
This is the rule: arn
Then I subtracted the sequence from the sequence multiplied by the common ratio .
This gives:
(r-1)S= -a +arn
= a(rn-1)
S=a(rn-1)
(r-1)