Notes 12.2: Geometric Sequences and Series - CVUSD Home

1 The table shows the heights of a bungee jumper s height of the bounces shown in the table above form a Geometric a Geometric sequence , the ratio of successive terms is the same number r, called the common What is a Geometric sequence ? Notes : Geometric Sequences and SeriesEx 1: Find the next three terms in the Geometric , 4, 16, 64,..Step 1 Find the value of rby dividing a term by the one before it. each term by the one before Finding Subsequent TermsStep 2 Multiply each term by 4 to find the next three 256 1024 4096 4 4 4 The next three terms are 256, 1024, and 2 1=41=4Ex 2: Find the next three terms in the Geometric 1 Find the value of r. = 2 1=3 9= 13 Step 2 Multiply each term by to find the next three next three terms are When the terms in a Geometric sequence alternate between positive and negative, the value of ris HintThe pattern in the table shows that to get the nth term, multiply the first termby the common ratioraised to the power (n 1).

2 The nth term of a Geometric sequence with common ratio r and first term a, isIII. Formula for the nth termEx 1: The first term of a Geometric sequence is 500, and the common ratio is What is the 7th term of the sequence ? an= a1rn 1 Write the 500( )7 1 Substitute 500 for a1,7 for n, and for 500( )6 Simplify the a 7th term of the sequence is 2: For a Geometric sequence , the first term is 5 and the common ratio is 2. Find the 6th term of the sequence ? an= a1rn 1 Write the 5(2)6 1 Substitute 5 for a1,6 for n, and 2 for 5(2)5 Simplify the 160 Use a 6th term of the sequence is 3: If the 9th term of ageometric sequence is 13,122 and the common ratio is -3, find the first term. an= a1rn 1 Write the ,122 = a1( 3)9 1 Substitute 13,122 for an, 9 for n, and 3 for ,122= a1 ( 3)8 Simplify the a first term of the sequence is ,122= a1 ( 3)813,122= a1 (6561)2= a1 Ex 4: What is the 8th term of the sequence 1000, 500, 250, 125.

3 ? an= a1rn 1 Write the the a 8th term of the sequence is 1000 for a1,8 for n, and for 1000( )8 1r= 2 1=5001000=12A ball is dropped from a tower. The table shows the heights of the ball s bounces, which form a Geometric sequence . What is the height of the 6th bounce?BounceHeight (cm)13002150375IV. Application of Geometric Sequencer= 2 1=150300=.5an= a1rn 1 Write the 300( )6 1 Substitute 300 for a1, 6 for n, and for 300( )5 Simplify the a height of the 6th bounce is 2: A population of fruit flies is growing in such a way that each generation is times as large as the last generation. Suppose there were 100 insects in the first generation. How many would there be in the fourth generation?SolutionThe populations form a Geometric sequence with a1= 100 and r=.

4 Use n = 4 in the formula for the fourth generation, the population is about 338 a1rn 1a4= 100( )4 1a4= 100( )3a4= Geometric means The terms between any two nonconsecutive terms of a Geometric sequence are called Geometric means. Ex 1. Write a sequence that has two Geometric means between 48 and -750. This sequence will have the form 48, ___, ___, -750. First, find the common a1rn-1a4= 48r4-1-750 = 48r3-125/8= r3 = r take cube root of both sides Now, determine the Geometric sequence a2= 48( ) a2= -120 a3= -120( ) a3= 300 sequence : 48, -120, 300, -750 Ex 2. Write a sequence that has two Geometric means between 128 and sequence will have the form 128, ___, ___, 54. First, find the common a1rn-1a4= 48r4-154= 128r327/64= r3.

5 75 = r take cube root of both sides Now, determine the Geometric sequence a2= 128(.75) a2= 96 a3= 96(.75) a3= 72 sequence : 128, 96, 72, 54V. Sum of a Finite Geometric Series The sum of the first n terms of a finite Geometric Series is given by = 1 1 1 Ex 1: find the sum of the first ten terms of the Geometric sequence 16-48+122-432+..Next, use the formula. 10= 1 1 1 10=16 16( 3)101 ( 3) 10=16 16( 3)101 ( 3)First, find the common ratio = 2 1 = 48 16r=-3 10=16 16(59049)1 ( 3) 10= 236,192 Ex 1: find the sum of the first eight terms of the Geometric sequence 14-70+350-1750+..Next, use the formula. 8= 1 1 1 8=14 14( 5)81 ( 5) 8=14 14(390626)6 First, find the common ratio = 2 1 = 70 14r=-5 8=14 54687506 8= 911,456VI.

6 Applications of Geometric Series On April 1 of every year for 25 years, Andrea deposits $2000 in an IRA which pays an APR of 10% compounded annually. If she makes no withdrawals, how much will she have in the account at the end of 25 years? = 1 1 1 =2200 2200( )251 =216, 1=2000 we can t start with zero (we would just get zero as our answer) we start at the end of the first 25 We are looking for the end of the because each year we had what we had before (100% = 1) plus our interest, so 1 + .1 =