Geometric sequences and Series

Geometric Sequence   and   Sum of a Geometric Series

RAFAY KHAN

 Overview

        Geometric Series and Sequence deals with patterns in a set of numbers. In this chapter, you will learn to find any number in a particular sequence; also, you will learn to find the sum of series of numbers, without adding them one-by-one. This chapter is requires; Patterns and Sequence identification skill, and solving equation skills. Here is a brief description of GEOMETRIC SEQUENCE and SUM OF A GEOMETRIC SERIES.    

Geometric Sequence:

In geometric sequence, the number obtained by dividing any term by the preceding term is a constant. This constant is the multiplier or the common ratio. Geometric Sequence and series describe exponential growth when the common ratio is greater then 1.

General Term of a Geometric Sequence:

Consider the geometric sequence 2, 6, 18,… Each term is 3 times the preceding term. We can write the term as follows.

To determine any term, we multiply the first term by a certain number of 3s. The number of 3s to multiply by is 1 less then the term number.

For example,

Similarly,

The term,, is the general term of the sequence.

We can use this general term to calculate any term in this sequence. For example, to determine the 8^th term,, substitute n=8 in .

The general term of Geometric Sequence is:

tn: General term

a: Initial value of sequence

r: common ratio

n: position of term in sequence

Sum of a Geometric Series

2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024

This expression indicates that the terms of geometric sequence are to be added. It is an example of a geometric series. A geometric series is the indicated sum of the terms of geometric sequence. To determine the sum of the series, add all 10 term. Instead of adding 10 terms, here is a more efficient formula you can use to add the 10 terms.

Let represent the sum of the first n term of the series.

Multiply by r:                                       

Subtract (1) from (2)

Factor:

Divide by r-1:

The general term for the sum of geometric series is:    

Sn: General term

a: Initial value of sequence

r: common ratio

n: position of term in sequence

Study Tips,  Methods  and or Advice

Tips

v     Always right the general term first before you solve the question

v     Determine the variables chart before solving the questions

Here is some of the stuff you should be careful about when solving the questions.

1)

 when solving the formula always keep in mind that the term “n-1” is solve first then multiply by the “r” term. Also we don’t multiply “r” term with “a” then with “n-1”.

Here are two examples:

5, 30, 180,…

    CORRECT                                                              WRONG

2)

If for example a sequence is given:  8, 4, 2, 1, then remember the common ratio is 0.5 or ½ multiply not divided by a factor of 2. In this sequence, the first term is multiply by  to get the second term.

 CORRECT      

                                                                               WRONG

Method To Solve Different Type Of Question

1)  If you get these types of question then have is an example of how you solve them. Insert two numbers between 5 and 320, so the four numbers form a geometric sequence.

Let r represent the common ratio.

To go from 1^st term to the 4^th term, multiply by r three times. Write an equation to represent this.

Since 4 x 4 x 4= 64

              r = 4

The geometric sequence is

Or 5, 20, 80, 320.

2) If the question is in fraction like  then look at the all denominators separately and all numerators separately. If you look each separately then you will notice the numerators are geometric series and denominators are arithmetic series. So in order to solve this type of question you use geometric sequence formula in numerator and arithmetic sequence formula in the denominator.

For example;  

If you have to determine the n^th  term of the sequence then you do the followings:

Ways To Solve Geometric Sequence:

There are two ways you can solve a geometric sequence. You can solve it either by making a chart or by simply using the general sequence formula form.

Sample question:

   The geometric sequence 2,6,18… is given determine the 7^th term in the sequence.

CHART METHOD:

If you use chart to determine the n^th  term of  a sequence then all you have to do is find the common ratio and keep multiplying until the n^th term.

GENERAL SEQUENCE FORMULA:

However, if you choose to use the formula then you won’t have to go through so much work. This formula is fast and simple. Here is an example of how you would solve the same question using the formula. 

Sample Questions and complete solutions

Extra Practice Questions and Answers

Self Reflection

RAFAY KHAN

Geometric series and sequences is one of the easiest chapters you will encounter in the Functions and Relations course. Although it’s easy, it’s tricky.  The trick to this chapter is to look for patterns in numbers. If you can’t see the patterns right away or you aren’t that good in finding patterns, then you might have some trouble in this chapter. With practice, you will be much more familiar with these patterns and you’ll be able to see the patterns right away.

This chapter is very useful for the chapters that follow because you become more familiar with patterns and numbers, which play a very important role in the upcoming chapters. What I like about this chapter is that it was understandable and it came handy in chapter 2. When I had to factor numbers, it was easy for me to tell if the numbers is a perfect square or not and it was easy for me to find the common factor, since I dealt with lot of numbers in this chapter.

The best advice I can give you for this chapter is to do the homework. Although it may seem easy at first, there are several surprises that you must be aware of so I highly stress that you take this chapter seriously. This webpage, which I have created, is very straight forward I hope you benefit from it and understand what I have explained.

BEST OF LUCK!!!!!