Sequences and patterns
Sequences and patterns arise naturally in many real life situations. Here is a stunning example to introduce the topic
Say for instance you go to the bank to deposit money and the bank gives you the following two options to choose from:
Option A
Deposit 1000 dollars.
The second day you receive 1100
The third day you receive 1200
The fourth day your receive 1300
And so forth....
Option B
Deposit 1 dollar.
The second day you receive 3
The third day you receive 9
The fourth day your receive 27
And so forth....
Which option gives you more money in 10 days?
At first, the tendency is say that option A is the best option
However, let us take a look and see what is going on here
If you choose option A,
The fifth day you receive 1400
The sixth day you receive 1500
The seventh day your receive 1600
The eighth day you receive 1700
The ninth day you receive 1800
The tenth day your receive 1900
On the other hand, if you choose option B,
The fifth day you receive 81
The sixth day you receive 243
The seventh day your receive 729
The eighth day you receive 2187
The ninth day you receive 6561
The tenth day your receive 19683
No doubt now you can see clearly that option B is the best option.
Notice that in option A, to get to the next number, just add 100 every time
We call this pattern an arithmetic sequence. To learn more about this type of sequence, go to
arithmetic sequence
In option B, to get to the next number, just multiply by 3 every time
We call this pattern a geometric sequence. To learn more about this type of sequence, go to
geometric sequence
The reason the money grew so fast in option B is because the pattern is an exponential growth, which usually grows fast.
Thw exponential growth above can be modeled with an exponential function
The exponential function is 3
^{n}
when n = 1, 3
^{1}= 3
when n = 2, 3
^{2}= 9
And so forth....