warm up section 3.6b (1). show that f(x) = 3x + 5 and g(x) = are inverses. (2). find the inverse of...

Warm Up Section 3.6B

(1). Show that f(x) = 3x + 5 and g(x) = are inverses.

(2). Find the inverse of h(x) = 8 – 3x.

(3). Solve: 27 x – 1 < 92x + 3

(4). Is k(x) = 4.6x increasing or decreasing throughout its domain?

x – 5 3

Answers to Warm Up Section 3.6B

(1). f(g(x)) g(f(x)) = f( ) = g(3x + 5)

= 3( ) + 5 =

= x – 5 + 5 =

= x = x

x – 5 3x – 5 3

3x + 5 – 5 3 3x 3

(2). h-1(x) =

(3). x > -9

(4). k(x) = 4.6x is increasing throughout its domain

x – 8 -3

Geometric Sequences and Series

Section 3.6

Standard: MM2A2 f g

Essential Question: What are the sums of finite geometric sequences and series? How do geometric sequences relate to exponential functions?

Let’s begin our lesson with a graphing reviewof exponential functions.

1) Graph y = 3x-1 +2.

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

x y

Growth

x y

0

1

1

333.2

111.2

211

53

3

Domain:

Range:

Asymptotes:

Zeros:

y-intercept:

Interval of increasing:

Interval of decreasing:

Rate of change (2 ≤ x ≤ 3)

End behavior:

All reals

y > 2

y = 2

None

(0, 2.33)

All reals

None

(2, 5) (3, 11)

x → -∞, y → 2; x → ∞, y → ∞

61

6

23

511

m

2) Graph

-8 -6 -4 -2 2 4 6 8

8

6

4

2

-2

-4

-6

-8

x y

Decay

42

13

x

y

45

320

275.3

5.3

3

1

Domain:

Range:

Asymptotes:

Zeros:

Y-intercept:

Interval of increasing:

Interval of decreasing:

Rate of change (-3 ≤ x ≤ -2)

End behavior:

All reals

y > -4

y = -4

(-5, 0)

(0, -.3875)

All reals

None

(-3, -3) (-2, -3.5)

x → -∞, y → ∞; x → ∞, y → -4

2

1

1

5.0

)3(2

)3(5.3

m

Vocabulary

Geometric Sequence: A sequence in which the ratio of any term to the previous term is constant.

Common Ratio: The constant ratio between consecutive terms of a geometric sequence, denoted by r.

Geometric Series: The expression formed by adding the terms of a geometric sequence.

Investigation 1: Recall: An arithmetic sequence is a sequence in which the difference between two consecutive terms is constant.

The constant difference between terms of an arithmetic sequence is denoted d and the explicit formula to find the nth term of a sequence is: an = a1 + d(n – 1).

1. Identify the next three terms of the arithmetic sequence, then write the explicit formula for the

sequence: 3, 7, 11, 15, an = 3 + 4(n – 1) or an = 4n – 1

2. Use the formula from example #1 to find the 27th term of the sequence.

a27 = 3 + 4(27 – 1) =

3. What is the sum of the first 27 terms of this sequence. Hint: use 1( )

2n n

nS a a

27

27(3 107) 1485

2S

19, 23, 27, . . .

107

In an arithmetic sequence, the terms are found by adding a constant amount to the preceding term. In a geometric sequence, the terms are found by multiplying each term after the first by a constant amount. This constant multiplier is called the common ratio and is denoted r.

For each geometric sequence, identify the common ratio, r.4. 2, 6, 18, 54, 162, . . .

5. 5, 50, 500, 5000, . . .

6. 3, , , , . . .

7. -4, 24, -144, 864, -5184, . . .

3

2

3

43

8

r = 3

r = 10

r = ½

r = -6

Tell whether the sequences is arithmetic, geometric or neither. For arithmetic sequences, give the common difference. For geometric sequences, give the common ratio.  8. 5, 10, 15, 20, 25, …. 9. 1, 1, 2, 3, 5, 8, 13, 21, …

10. 1, -4, 16, -64, 256, …

11. 512, 256, 128, 64, 32, …

arithmetic; d = 5

neither

geometric; r = -4

geometric; r = ½

Check for Understanding:  12. Find the first four terms of a geometric sequence in which a1 = 5 and r = -3.

_____ , _____ , _____ , _____.  

13. Find the missing term in the geometric sequence: -7, _______ , -28, 56, _______ , . . .

5 -15 45 -135

× -3 × -3 × -3

56 ÷ -28 = -2So, r = -2

× -2

14

× -2

-112

Investigation 2: The explicit formula used to find the nth term of a geometric sequence with the first term a1 and the common ratio r is given by: an = a1∙ rn-1

Write a rule for the nth term of the sequence given. Then find a10.  14. 972, -324, 108, -36, … Rule: an = 972∙(-⅓)n-1

a10 = 972∙(-⅓)10-1 = ____  

4

81

15. 1, 6, 36, 216, 1296, …

Rule: an = 1∙6n-1

a10 = 1∙610-1 = 10077696

  16. 14, 28, 56, 112, …

Rule: an = 14∙2n-1

a10 = 14∙210-1 = 7168

Check for Understanding: 17. If a5 = 324 and r = -3, write the explicit formula for the geometric sequence and find a10.

_____ , _____ , _____ , _____, 324 

Rule: an = 4∙(-3)n-1

a10 = 4∙(-3)10-1 = -78732 

÷ -3

-108

÷ -3

36

÷ -3

-12

÷ -3

4

5 1

1324 3a

1324 81a

14 a

OR

18. If a3 = 18 and r = 3 write the explicit formula for the geometric sequence and find a10.

Rule: an = 2∙(3)n-1

a10 = 2∙(3)10-1 = 39366

3 1

118 3a

118 9a

12 a

19. If a3 = 56 and a6 = 448 complete the following for the geometric sequence: 

_____, _____, 56, _____, _____, 448  

4 1448 56 r

38 r

2 r

÷ 2 ÷ 2

2814

× 2 × 2

112 224

20. If r = 2 and a1 = 1 for a geometric sequence, a. Write a rule for the nth term of the sequence.

b. Graph the first five terms of the sequence. (1, 1), (2, 2), (3, 4), (4, 8), (5, 16)

c. What kind of graph does this represent? exponential

11 2

n

na