3.3: Recognising Sequences
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For some of the sequences, we can predict the nth term easily, and the explicit (general) formula can be checked using induction. In this section, we will explore this kind of sequence.
Example 3.3.1: Harmonic sequence
Find a formula for the nth term of the a sequence that has the following initial terms: 12,13,14,15,16,⋯.
- Answer
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1n+1
Caution
Two sequences may start with the same initial terms but diverge later on.
Example 3.3.2: One's
1×11=11
11×11=121
111×111=12321
1111×1111=1234321
Can you predict the pattern?
Example 3.3.3: Perfect Squares
Find a formula for the nth term of the following sequence 1,4,9,16,25,⋯.
- Answer
-
n2.
Here is another way to find the nth term:
Note that the difference in the sequence 1,4,9,16,25,⋯ is 3,5,7,9,⋯ and the difference in this sequence is 2. Therefore the nth can be written as a quadratic formulae pn2+qn+c. p,q,c can be found using the following information:
tn | dn=tn−t(n−1) | d=dn−d(n−1) | |
---|---|---|---|
n=1 | p+q+c | ||
n=2 | 4p+2q+c | 3p+q | |
n=3 |
9p+3q+c | 5p+q | 2p |
n=4 | 16p+4q+c | 7p+q | 2p |
Hence, in our case, 2p=2. Therefore, p=1. Now 3p+q=3 and p=2. Thus q=0. Now p+q+c=1, which implies c=0. Hence, tn=n2.
Quadratic Sequences:
A sequence is called a quadratic sequence if the differences of consecutive terms, dn=tn−t(n−1) differ by the same amount d=dn−d(n−1),∀n∈N. In this case, the nth term of the sequence is given by
tn=a+(n−1)d1+(n−1)(n−2)d2,
where d1 is the first difference between the first and second terms of the sequence, and d is the common second difference. This result can be shown by using induction. We will explore this in the next section.
Below is another way to solve this:
Example 3.3.4: Quadratic Sequences
Find a formula for the nth term of the following sequence: 2,6,12,20,⋯
261220
468
22
Hence the nth term has a term (2/2)n2.
tn | 2 | 6 | 12 | 20 |
n2 | 1 | 4 | 9 | 16 |
n2−tn | 1 | 2 | 3 | 4 |
Notice that n2−tn is an arithmetic sequence with the first term 1 and the common difference is 1. Thus n2−tn=1+1(n−1). Hence tn=n2−n.
Triangular numbers
Thinking Out Loud:
1,3,6,10,15,21,28,... are triangular numbers: you can arrange them in a triangular array:
Tn=nth triangular number. What is the hundredth triangular number?
Example 3.3.5: Hexagonal Tilling (Centered hexagonal numbers)
Find the nth term of the sequence 1,7,19,37,⋯. There are 6 triangular numbers and 1 center; therefore, 6(n−1)n2+1=3n(n−1)+1=3n2−3n+1.
6(n−1)n2+1=3(n(n−1))+1=3n2−3n+1.
Example 3.3.6:
Consider the sequence of tilling using hexagons. The hexagon numbers are the sequence 1,6,15,···. Predict the nth term. Explain your prediction.
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|
|
|
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A |
Term # (n) |
1 |
2 |
3 |
4 |
B |
# of hexagons |
1 |
6 |
15 |
28 |
C |
BA |
1 |
3 |
5 |
7 |
D |
2(n)−1 |
1 |
3 |
5 |
7 |
Since we know that (A)(C)=B, we can conclude that the term number (A)=n, multiplied by C(2n−1) will give you the corresponding number in the sequence B.
Therefore: tn=n(2n−1)=2n2−n.
Tower of Hanoi
According to the legend of the Tower of Hanoi (formerly the "Tower of Brahma" in a temple in the Indian city of Benares), the temple priests are to transfer a tower consisting of 64 fragile disks of gold from one part of the temple to another, one disk at a time. The disks are arranged in order, no two of them the same size, with the largest on the bottom and the smallest on top. Because of their fragility, a larger disk may never be placed on a smaller one, and there is only one intermediate location where disks can be temporarily placed. It is said that before the priests complete their task the temple will crumble into dust and the world will vanish in a clap of thunder.
Let A, B and C be the posts. Then •1 disk: 1 move
Move 1: move disk 1 to post C
•2 disks: 3 moves
Move 1: move disk 2 to post B
Move 2: move disk 1 to post C
Move 3: move disk 2 to post C
•3 disks: 7 moves
Move 1: move disk 3 to post C
Move 2: move disk 2 to post B
Move 3: move disk 3 to post B
Move 4: move disk 1 to post C
Move 5: move disk 3 to post A
Move 6: move disk 2 to post C
Move 7: move disk 3 to post C
The number of moves needed to transfer n disks from post-A to post C is 2M+1, where M is the number of moves needed to transfer n−1 disks from post A to post C. This is called a recursive sequence. Can we able to guess a formula depending on n only?
Number of Disks | Min. number of Moves |
1 |
1 |
2 | 3 |
3 | 7 |
4 | 15 |
5 | 31 |
From this pattern, we can guess the formula for finding the minimum number of moves it takes to transfer n disks from post-A to post C is: 2n−1. This guess can be proved by using induction.
Recursive Sequences
Definition
A recurrence relation for a sequence {an} is formula that relates to each term an to its predecessors a0,a1,⋯,an−1.
Thinking Out Loud:
A tiling covers a region using tile pieces from some given set to cover the region without overlaps. How many ways can you arrange the 2×1 dominoes to cover the 2×n checkerboard?
Fibonacci Sequences
Fibonacci sequence: 1,1,2,3,5,8,13,21,34,55,,...
Let Fn=nth Fibonacci Number.The n term in the Fibonacci sequence is obtained by adding the previous two terms.
That is, Fn=F(n−1) +F(n−2), n>2,F1=1, and F2=1
Some facts about the Fibonacci sequence :
- The sum of the first n even-numbered Fibonacci numbers is one less than the next Fibonacci number.
- The only square Fibonacci numbers are 0,1 and 144.
- The sum of the first n odd-numbered Fibonacci numbers is the next Fibonacci number.
Example 3.3.9:
Consider the Fibonacci sequence: 1,1,2,3,5,8,13,21,34,55,,...
The squares of the Fibonacci sequence: 1,1,4,9,25,64,169..... Consider the sum of the squares of consecutive terms, F2(n−1) +F2(n−2), for n>2. That is:
1+1=2
1+4=5
4+9=13
What can you say about the resulting number?
- Answer
-
It is Fibonacci. In fact, F(2n−3)=F2(n−1) +F2(n−2).
Example 3.3.11: More Fibonacci .jpg?revision=1&size=bestfit&width=356&height=221)
Let an be the Fibonacci sequence. That is an=an−1+an−2,n∈N, with a1=1,a0=0, where N be the set of all natural numbers.
Consider the Fibonacci squares illustrated in the figure:
Let hn be the shortest (perpendicular) distance between nth parallel diagonal vectors. Consider the figure:
Comparing the area of trapezoids, we get
hn=a2n−a2n−1√2(an−an−1)=1√2(an+an−1).
Hence, hn is a Fibonacci-like sequence.
- Tower of Hanoi by https://texample.net/tikz/examples/towers-of-hanoi/