Section 1-7Explicit Formulas for Sequences

In-Class Activity

See page 41.

Term:

Term:

Each figure, set of numbers, or value

Term:

Each figure, set of numbers, or value

Explicit Formula for the nth Term:

Term:

Each figure, set of numbers, or value

Explicit Formula for the nth Term:

Allows for us to find any term in a sequence

Term:

Each figure, set of numbers, or value

Explicit Formula for the nth Term:

Allows for us to find any term in a sequence

Sequence:

Term:

Each figure, set of numbers, or value

Explicit Formula for the nth Term:

Allows for us to find any term in a sequence

Sequence:

A function whose domain is the natural numbers

Example 1Use the formula we derived in the activity to find t15.

Example 1Use the formula we derived in the activity to find t15.

tn= n(n +1), for int. n ≥1

Example 1Use the formula we derived in the activity to find t15.

tn= n(n +1), for int. n ≥1

t15=15(15 +1)

Example 1Use the formula we derived in the activity to find t15.

tn= n(n +1), for int. n ≥1

t15=15(15 +1)

=15(16)

Example 1Use the formula we derived in the activity to find t15.

tn= n(n +1), for int. n ≥1

t15=15(15 +1)

=15(16)

= 240

Subscript/Index

Subscript/Index

Tells us which term in the sequence that is being dealt with

Subscript/Index

Tells us which term in the sequence that is being dealt with

t15 is the 15th term in the sequence

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

t3 = 7(3) + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

t3 = 7(3) + 1 = 21 + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

t3 = 7(3) + 1 = 21 + 1 = 22

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

t3 = 7(3) + 1 = 21 + 1 = 22

t4 = 7(4) + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

t3 = 7(3) + 1 = 21 + 1 = 22

t4 = 7(4) + 1 = 28 + 1

Example 2

a. Find the first 4 terms. tn= 7n +1, for int. n ≥1.

t1 = 7(1) + 1 = 7 + 1 = 8

t2 = 7(2) + 1 = 14 + 1 = 15

t3 = 7(3) + 1 = 21 + 1 = 22

t4 = 7(4) + 1 = 28 + 1 = 29

Example 2

b. Find t12 and state what it means. tn= 7n +1, for int. n ≥1.

Example 2

b. Find t12 and state what it means. tn= 7n +1, for int. n ≥1.

t12 = 7(12) + 1

Example 2

b. Find t12 and state what it means. tn= 7n +1, for int. n ≥1.

t12 = 7(12) + 1 = 84 + 1

Example 2

b. Find t12 and state what it means. tn= 7n +1, for int. n ≥1.

t12 = 7(12) + 1 = 84 + 1 = 85

Example 2

b. Find t12 and state what it means.

The twelfth term of the sequence is 85.

tn= 7n +1, for int. n ≥1.

t12 = 7(12) + 1 = 84 + 1 = 85

Example 3

Matt Mitarnowski is standing on the top of an 80 foot-high wall. Don’t ask me why he’s up there. He’s weird like that. But while he’s up there, he

decided to drop a ball, which will bounce back 70% of its previous height.

Example 3a. Write an explicit formula for this situation

Example 3a. Write an explicit formula for this situation

b

n= 80(.7)n

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1 b

n= 80(.7)n

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

= 39.2

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

= 39.2

b

3= 80(.7)3

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

= 39.2

b

3= 80(.7)3

= 27.44

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

= 39.2

b

3= 80(.7)3

= 27.44

b

4= 80(.7)4

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

= 39.2

b

3= 80(.7)3

= 27.44

b

4= 80(.7)4

=19.208

Example 3a. Write an explicit formula for this situation

for all int. n ≥ 1

b. Write the first four terms of this sequence b

n= 80(.7)n

b

1= 80(.7)1 = 56

b

2= 80(.7)2

= 39.2

b

3= 80(.7)3

= 27.44

b

4= 80(.7)4

=19.208

Don’t forget the labels! All answers are in feet.

Example 3c. After how many bounces will the ball bounce less

than 9 feet?

Example 3c. After how many bounces will the ball bounce less

than 9 feet?

b5 = 13.4456

Example 3c. After how many bounces will the ball bounce less

than 9 feet?

b5 = 13.4456

b6 = 9.41192

Example 3c. After how many bounces will the ball bounce less

than 9 feet?

b5 = 13.4456

b6 = 9.41192

b7 = 6.588344

Example 3c. After how many bounces will the ball bounce less

than 9 feet?

It will take 7 bounces until the ball bounces less than 9 feet.

b5 = 13.4456

b6 = 9.41192

b7 = 6.588344

Homework

Homework

p. 45 #1-27