Geometric Sequences & Series

Sorry it's taken me so long to post this blog entry!! Lets recap what has happened over the past couple of days..

Hopefully everyone was happy with their test results, had a great weekend, and finished the assignment that was due. Also, Happy belated "name day" to Mr. P's mom!
Oh yeah! We also stared learning a new unit.

Geometric Sequences & Series:

Lets start with the basics;
Example: Find the next term in each sequence. Describe how you got them.

1, 4, 9, 16, .... The answer is 25. We get this by squaring the term numbers and then following the pattern that they make. Which in this case is 1, 2, 3, 4, 5, ...
Eg) 1^2=1, 2^2=4, 3^2=9, 4^2=16... so 5^2=25

5, 8, 11, 14, ... The answer is 17. We get this by adding 3 to the previous number.
Eg) 5+3=8, 8+3=11, 11+3=14... so 14+3=17

2, 6, 18, .... The answer is 54. We get this answer by multiplying the previous number by 3.
Eg) 2x3=6, 6x3=18... so 18x3=54

The second Sequence is something called an Arithmetic Sequence.

Terms: t1 t2 t3 t4

Sequences: 5 5+(3) 5+2(3) 5+3(3)

How would you find the 100th term?

To find the 100th term we have to follow the rules that we just discussed;

t100= 5+99(3)

t100=302


Arithmetic Sequences:

An Arithmetic Sequence is a sequence like: 6, 10, 14, 18 ... in which each term can be calculated by adding a consistent value to the preceding term. The value is called the common difference, d.

There are two definitions/equations that you need to know for this type of Sequence:

*Recursive definition: tn=tn-1+d (you must know the previous term to find the next term)

*Explicit definition: tn=t1+(n-1)(d) Where:

d is the common difference
t1 is the value of the first term
n is the number of terms
tn is the value of the nth term


Example) Given the Arithmetic Sequence: 5, 13, 21, ...

A) Write a function to generate the Sequence:

tn=t1+(n-1)(d)

tn=5+(n-1)(8)

tn=5+8n-8

tn=8n-3

B) Find the value of the 18th term.

t18=8(18)-3

t18=141


Example 2) Given that an Arithmetic Sequence has t5=6 and t8=27, find term 2.

first of all, we know that there is always a difference between terms. So if we have term 5 and term 8, there will be three differences between them. We can
represent this as 3d.
So...
6+3d=27

3d=21

d=7

There is also a three number space between term 5 and term 2.
Therefore...
Term 2=6-3d

6-21=-15

t2=-15

Hopefully this helps!!!