1

6.2 Geometric Sequences

A geometric sequence is a sequence in which the ratio between a term and its preceding term is a constant.

The number, r, is called the common ratio for the geometric sequence. For the geometric sequence above, the common ratio is 2.

; 2 510

aa

1

2 ; 2 1020

aa

2

3 . . . ; 2 2040

aa

3

4 . . . ; 2 320640

aa

7

8 . . . ; 2 512010240

aa

11

12

For a geometric sequence, , for every positive integer k.r a

a

k

1k

For example; 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, . . .

In the previous section, we learned about arithmetic sequences. Arithmetic sequences have a common difference. Example: 3, 8, 13, 18, 23, 28, . . . (Arithmetic Sequence)Another type of sequence is a geometric sequence. The ratio of a term in a geometric sequence to its preceding term is always the same number.

For a geometric sequence, the general term is: an = a1 r(n-1) ,where a1 is the first term and r is the common

ratio.

General Term of an Arithmetic Sequence

2

6.2 Geometric Sequences

Example 1. Find the general term for the geometric sequence: 8, 32, 128, 512, 2048, . . .

The general term for that in a geometric sequence, an = a1 r(n-1)

So for the geometric sequence above, an = (8) 4(n-

1)

The common ratio is,

Note that we could have used the ratio between any two consecutive terms, i.e.,

2

1

a 324

a 8

5

4

a 20484

a 512

Simplifying this expression even more: (n 1)3 2na 2 2

3 2n 2na 2 2

3 2n 2na 2

2n 1nAnswer: a 2

Your Turn Problem #1

1

4, 2,1, ,2

Find the general term for the geometric sequence:

n 3nAnswer: a 2

3

6.2 Geometric Sequences

Find the general term for the geometric sequence: 3Example 2. , 9, 27, 81,

The common ratio, 9

r 33

Now we will use the general term formula of a geometric sequence, n 1n 1a a r

n 1

na 3 3

1 n 1

na 3 3

nnAnswer: a 3

Your Turn Problem #2

0.2, 0.04, 0.008, 0.0016,…Find the general term for the geometric sequence:

n

nAnswer: a 0.2

4

6.2 Geometric Sequences

For this sequence, the common ratio, r = (-24) (-4) = 6

Example 3. Find the 9th term for the geometric sequence, -4, -24, -144, -864, . . .

For a geometric sequence, the formula for the general term can be used to find a desired term.

So, an = (-4) 6(n–1)

Thus, a9 = (-4) 6(9–1)

= (-4) 68

= (-4) (1,679,616)= -6,718,464

Answer: The 9th term, a9 = -6,718,464

Find the 12th term for the geometric sequence: 1.3, 3.9, 11.7, 35.1, . . .

Your Turn Problem #3

Answer: The 12th term, a12 = 230,291.1

Since we want the 9th term, let n = 9.

5

6.2 Geometric Sequences

To find the sum of an geometric sequence, we need to know the first term, a1, the common ratio, r, and the number of terms, n.

Adding the Terms of a Geometric Sequence

n1 1

n

ar aS

r 1

n1

n

a 1 rS

1 r

or

Note: There are many forms of the formula for the sum of a geometric sequence. The two forms above are commonly used.

Sum formula for the first n terms of a geometric sequence.

Example 4. Find the sum of the first 8 terms of the geometric sequence, 6, 60, 600, 6000, . . . For this sum, n = 8 and r = 60 6 = 10

Using the first form of the finite sum formula with a1 = 6, n = 8, and r = 10, we get

Answer: The sum of the first 8 terms is 66,666,666.

8

8 6 1 10

1 10S

6 99,999,999

-9 66,666,666

Your Turn Problem #4

Find the sum of the first 12 terms of the geometric sequence, 3, -12, 48, -192, . . .

Answer: The sum of the first 12 terms is S12 = -10,066,329

6

6.2 Geometric Sequences

For this sequence, r = -175 875 = -1/5 For this sum, n = 9

Example 5. Find the sum of the first 9 terms of the geometric sequence, 875, -175, 35, 7, . . .

The finite sum formula can be used with any geometric sequence even if the common ratio, r, is a fraction or a negative number.

Using the finite sum formula with a1 = 875, n = 9, and r = -1/5, we get

9

9

875 1

1

1

5

1

5

S

Answer: The sum of the first 9 terms,

S9 = 729 522/3125 or 729.16704

522 729 or 729.16704

3125

875

6

5

11

1953125

19531261953125

65

875

Your Turn Problem #5

Find the sum of the first 8 terms of the geometric sequence, 144, 96, 64, 42 2/3, . . .

8

100880 35Answer: The sum of the first 8 terms is S or 415

243 243

7

6.2 Geometric Sequences

Example 6. Find 3 + 33 + 363 + 3,993 + . . . + 643,076,643

For a geometric sequence, if the number of terms of to be added is unknown, we need to either count the number of terms or use the nth-term formula to find n.

For this sum, we must first determine if this is a geometric sequence. To do this, determine if there is a common ratio between the terms.For this sequence, a2 a1 = 11, a3 a2 = 11, a4 a3 = 11. So, there is a common ratio, r = 11, which means that we are, in fact, working with a geometric sequence.

To find the number of terms, n, use the nth term formula. Then use the sum formula with n, a1 = 3, and r = 11,

n 1n 1a a r

n 1643,076,643 3 11

n 1214,358,881 11

(Solve by matching bases or using logs.)

n 1811 11

n 9

n1

n

a 1 rS

1 r

Now use the sum formula:

Answer: 707,384,307

9

9

(3) 1 11

(1 11)S

10-691)2,357,947, -(3)(1

Your Turn Problem #6

Find (-7) + (-14) + (-28) + (-56) + . . . + (-917,504)

Answer: r = 2 and n = 18. The sum is -1,835,001.

8

6.2 Geometric Sequences

Geometric Series

Summation notation can be used to indicate the sum of a geometric sequence. When asked to evaluate a sum, write out at least the first three terms. Good idea but not necessary to calculate the last term, n.

Example 7. Find

9i5

1 i

9

9

5(1 5 )S

(1 5)

We can calculate this sum using the finite sum formula with a1 = 5, r = 5, and n = 9. Recall n= 9 - 1+1 = 9. (upper limit – lower limit +1)

i 1 2 3 9

1

5 5 5 5 5

9

i

5 25 125 1953125

5(1 1,953,125)4

4-

9,765,620-

Answer: 2,441,405

Answer: a1 = -6, n = 12, and r = -6. The sum is 1,865,813,430.

Your Turn Problem #7

1

12

Find 6

i

i

9

6.2 Geometric Sequences

Example 8. Find i

i

13

(6 )

3

313

(6 ) (6 3) (6 4) (6 5) (6 13)

3

3 3 3 3 3

i

i

(3) (2) (1) ( 7)3 3 3 3

127 9 3

2187

11

11

127 1

3S

11

3

Answer:88,573 1,093

or 402,187 2,187

127 1

17714723

177,146 3 272177,147

Using the finite sum formula, n

1

n

a 1 rS

1 r

with a1 = 27, r = 1/3, and n = 11 (13-3+1), we get:

Answer: The sum is –531,684.

Your Turn Problem #8

5

11

Find 4 3

i

i

10

6.2 Geometric Sequences

1 2 3

1

3 3 3 3

4 4 4 4

i

i

3 9 27

4 16 64

Since |r|<1, we can use the infinite sum formula with a1 = ¾ and r = ¾ .

3

4S

31

4

3

43

1

4

1

3Answer: 3

4

i

i

Infinite Geometric Series

The sum of an infinite geometric sequence is give by the formula: Example 9. Find

1

3

4

i

i

1aS , for r 1

1 r

Your Turn Problem #9

1

i5Find -8

i 5

Answer: The sum is .13

11

6.2 Geometric Sequences

Using the infinite sum formula with a1 = 0.63 and r = 0.01 , we get

01.010.63

. . . 0.000063 0.0063 0.63 S

Writing Repeating Decimals in Fractional FormA repeating decimal number can be written as the sum of terms from a geometric sequence. The common ratio, r, is 10-R, where R equals the number of digits that are repeated. Since 0<| r|<1, the infinite sum formula can be used to write the repeating decimal in fractional form.

Write 63 as a mixed number whose fraction is in lExample owest terms. 10. 4.

Note that the terms 0.63, 0.0063, 0.000063, . . . form a geometric sequence with a common ratio of r = 10 -2 = 1/100 = 0.01 (two digits, 6 and 3, are repeated so R=2).

. . . 0.000063 0.0063 0.63 4 634.

990.0.63

63 7

or 99 11

7 7So, 4.63 4 4

11 11

268Answer: 2

333

Your Turn Problem #10

Write 2804 as a mixed number whose fraction is in lowest terms..

12

6.2 Geometric Sequences

Using the infinite sum formula with a1 = 0.057 and r = 0.01 , we get

0.057 0.057 0.00057 0.0000057 . . .

1 0.01S

Note that the terms 0.057, 0.00057, 0.0000057, . . . form a geometric sequence with a common ratio of r = 10 -2 = 1/100 = 0.01 (two digits, 5 and 7, are repeated so R=2).

. . . 0.0000057 0.00057 0.057 0.2 19 5719.2

0.057

0.99 57 19

or990 330

2 19So, 19.257 19

10 330 17

Answer: 19.257 1966

If part of a repeating decimal number does not repeat, then treat that non-repeating part as a terminating decimal when converting to fraction form.

Write 19 57 as a mixed number whose fraction is in lExample owest terms. 11. .2

34Answer: 42

55

Your Turn Problem #11

Write 42618 as a mixed number whose fraction is in lowest terms..

The End4-3-2007