# 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.

The exponential growth above can be modeled with an exponential function

The exponential function is 3n

when n = 1, 31= 3

when n = 2, 32= 9

And so forth....

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