Pre-Algebra Combo with "Math Lesson 69 Patterns and Functions: Sequences" and 4 others

Series

1+2+3+4+5+6+7+8+9+10

Find the next three terms in the sequence: 3, 7, 11, 15, ...

19, 23, 27

What is one way to find the limit of a sequence?

A coordinate graph (graphing the sequence)

Arithmetic Sequence

A list of numbers where you are adding or subtracting constantly by the same number

There are 5 polyhedra; Which one has 6 square faces?

Hexahedron

If the sequence of partial sum of an infinite series has a limit, then that limit is...

the sum of the series

Find the next three terms in the sequence: 4, -1, -6, ...

-11, -16, -21

Find the 12th term in a sequence with common difference -8 and first term 58.

-30

Find the 21st term in the sequence. -4, 1, 6, 11, 16, ...

-4 + (21-1)(5) -4 + 20 x 5 -4 + 100 = 96

Find the common difference in the sequence: 10, 3, -4, -11, ...

-7

Warm Up. What number should be added to the first number to get the second number? 1) 8 1/2, 10 2) 9, 12 1/2 3) 1 2/3, 2 1/3 4) 7 3/4, 9 1/4

1) 8 1/2, 10 (1 1/2) 2) 9, 12 1/2 (3 1/2) 3) 1 2/3, 2 1/3 (2/3) 4) 7 3/4, 9 1/4 (1 1/2)

Sequence

1,2,3,4,5,6,7,8,9,10

Find the nth term: -12, -2, 8, 18, ...

10n - 22

Extend a sequence by adding. Describe the pattern in the sequence 16, 24, 32, 40, ...

16, 24, 32, 40, ... Starting with 16 add 8 repeatedly.

Find the first term in a sequence where the common difference is 6 and the eleventh term is 77.

17

a₁

1st Term of a Sequence

Describe the pattern in the sequence 20, 16, 12, 8, ...

20, 16, 12, 8, ... Starting with 20 subtract 4 repeatedly.

Find the 10th term in the sequence: -5, -2, 1, 4, ...

22

Find the 72nd term in the sequence: -5, -1, 3, 7, ..

279

Find the nth term in the sequence. (Write the explicit formula): 10, 12, 14, 16, ...

2n + 8

Extend a sequence by multiplying. Describe the pattern in the sequence 5, 15, 45, 135, .....

5, 15, 45, 135, ..... Starting with 5 multiply by 3 repeatedly.

The NCAA basketball tournament starts with 64 teams. The 2nd round consists of 32 teams and the 3rd round consists of 16 teams. How many teams are in the fifth round?

64, 32, 16, 8, ____ ÷2 ÷2 ÷2 ÷2 The answer is 4.

Find the common difference: 16, 25, 34, ..

9

Find the 21st term in the sequence. -4, 1, 6, 11, 16, ...

96

Geometric Sequence

A sequence or list of numbers were you constantly multiply or divide by a common ratio

What is the difference between a finite sequence and infinite sequence?

An infinite sequence never ends, a finite sequence has an end.

There are 5 polyhedra; Which one has 12 pentagonal faces?

Dodecahedron

Explain how the numbers are related in the sequence 9, 3, 1, 1/3.

Each number is 1/3 of the number before it.

What is the formula and way of solving an arithmetic series?

First, you must use the formula aᴺ=a¹+d(n-1) to find the number of terms in the series, next you must take that number of terms and plug it into the formula: sᴺ=n(a¹+aᴺ)/2 (both the sᴺ and n are representing the number of terms which were previously found)

James borrowed $315 from his parents for a snowboard. he agreed to pay them back in monthly payments. In February he owed $265 after making his first payment. in March he owed $215. In April he owed $165. What are his monthly payments? How much will he owe in July?

He will owe $15 in July and his monthly payments are $50.

There are 5 polyhedra; Which one has 20 triangular faces?

Icosahedron

describe how a number sequence would be helpful in a contest or competition.

It would be useful in a contest or competition because you could figure out what place they can in.

There are 5 polyhedra; Which one has 8 triangular faces?

Octahedron

When is a˳ sometimes used?

To express a beginning value (starting point)

What is an arithmetic sequence?

When you add or subtract a certain(constant) amount to get to the next term (d- common difference, used to represent this constant number)

What is a geometric sequence?

When you multiply or divide a certain(constant) amount to get to the next term (r- common ratio, used to represent this constant number)

How do you solve for problems with subscripts? (asking to find the 3rd and 12th term)

You must plug in the number(which is subscript) for n in the equation and solve (3 plugged in for n and 12 plugged in for 12)

How do you use recursive formulas?

You use recursive formulas to find the nth term by using the term before it

arithmetic sequence

a numerical pattern that adds or subtracts at a constant rate or value

When the terms of a sequence are added together, what is the sum of the terms called?

a series

consecutive terms

a term directly after another term

Describe the pattern and give the next two numbers in the sequence. 8, 14, 20, 26, ... a. add 5; 32, 47 b. add 6; 32, 38 c. add 6, 32, 41 d. add 7; 34, 38

b. add 6; 32, 38

How could you find the limit of the partial sum of an infinite series?

you could graph the series to see which number its going towards if theres a limit

Patterns and Functions: Sequences

A sequence is a list of numbers in a specific order. by determining the pattern, you can find additional numbers in the sequence. Sequences are formed by adding and subtracting numbers called arithmetic sequences. Sequences formed by multiplying or dividing numbers are called geometric sequences.

sequence

An ordered list of numbers.

Drake is right because Meghan didn't add the 1 1/2, she only added 1.

Meghan and Drake are finding the missing number in the sequence 3, 4 1/2, ____, 7 1/2...Who is correct?

What does "compounded quarterly" mean for investment word problems?

Payment received four times per year

When you are solving for sigma notation problems and they are in the form of a geometric sequence, then you can use the formula ... ?

S=a¹-aᴺr/1-r (aᴺ is the last term of the series)

What does "Sigma" mean

Sigma stands for- summation

There are 5 polyhedra; Which one is a triangular pyramid? (4 faces)

Tetrahedron

Common Difference

The difference between a number and the next number in an arithmetic sequence:"d"

Common Ratio

The number that is constantly multiplied or divided by in a Geometric Sequence:"r"

Find the missing number. _____, 25, 5, 1 a. 125 b. 126 c. 252 d. 250

a. 125

Find the missing number. 39, 32 1/2, ____, 19 1/2,... a. 26 b. 25 c. 26 1/2 d. 25 1/2

a. 26

Describe the pattern, then find the next two numbers. 6, 11, 16, 21, ... a. add 5; 26, 31 b. add 5; 26, 34 c. add 4; 26, 39 d. add 6; 28, 31

a. add 5; 26, 31

Describe the pattern and give the next two numbers in the sequence. 3, 6, 12, 24, ... a. multiply by 2; 48, 96 b. multiply by 3; 96, 96 c. multiply by 1; 48, 384 d. multiply by 2; 48, 192

a. multiply by 2; 48, 96

What form does arithmetic sequences follow?

a¹+d(n-1)

What formula do you use to find the number of terms in a finite arithmetic sequence?

aᴺ=a¹+d(n-1) (n is the final term in aᴺ)

What form does geometric sequences follow?

aᴺ=a¹rᴺ-¹

Equation for Recursive Definition

aⁿ=aⁿ⁻₁+d

Arithmetic Equation: Finding the nth term

aⁿ=a₁+d(n-1)

Geometric Sequence: Finding the nth term

aⁿ=a₁∙rⁿ⁻¹

Find the missing number. _____, 32, 8, 2 a. 129 b. 128 c. 258 d. 256

b. 128

The first four terms in a pattern are 4, 7, 10, and 13. What is the fifth term in the pattern? a. 15 b. 16 c. 17 d. 18

b. 16

Thomas is getting in shape for track. he's starting with a 2 mile run and will increase the run by 1/2 mile each week for 4 weeks. What will his distance be for the second, third and fourth weeks? a. 3 mi, 4 1/2 mi, 6 mi b. 2 1/2 mi, 3 mi, 3 1/2 mi c. 4 mi, 5 1/2 mi, 7 mi d. 2 mi, 2 1/2 mi, 3 mi

b. 2 1/2 mi, 3 mi, 3 1/2 mi

Describe the pattern and give the next two numbers in the sequence. 5, 25, 125, 625, ... a. multiply by 6; 6,250; 15, 625 b. multiply by 5; 3,125; 15, 625 c. multiply by 4; 3,125; 156,250 d. multiply by 5; 3,125; 31,250

b. multiply by 5; 3,125; 15,625

Paco bought 10 feet of rope. he cut it into several 5/6 foot pieces. Which equation can you use to find how many pieces of rope Paco cut? a. 5/6 ÷ 10 = x b. 5/6 ÷ x = 10 c. 10 ÷ x = 5/6 d. 10x = 5/6

c. 10 ÷ x = 5/6

if the first four term,s in a number pattern are 3, 7, 12, and 18, what are the next two terms in the pattern? a. 22, 26 b. 23, 28 c. 25, 33 d. 24, 31

c. 25, 33

Find the missing number. _____, 25, 125, 625 a. 12 b. 10 c. 5 d. 6

c. 5

A pattern begins with the number 7 and repeatedly adds 5. What are the first four terms in this pattern? a. 7, 12, 19, 24 b. 5, 12, 19, 26 c. 7, 12, 17, 22 d. 7, 14, 21, 28

c. 7, 12, 19, 22

Find the missing number. ______, 27, 9, 3 a. 164 b. 162 c. 81 d. 82

c. 81

Find the missing number. ____, 64, 16, 4 a. 512 b. 514 c. 257 d. 256

d. 256

Solve by first solving a simpler problem. You and your friends are going to bake 48 loaves of bread for the bake sale. You need 1 1/4 cups of flour for each loaf. A bag contains 18 3/4 cups. How many bags of flour do you need for all of the loaves? a. 4 1/2 bags b. 4 1/4 bags c. 3 1/2 bags d. 3 1/5 bags

d. 3 1/5 bags

Solve by first solving a simpler problem. On a school day, jose spends 5 1/4 hours in class. Each class lasts 3/4 hour. How many classes does he have? a. 4 b. 5 c. 6 d. 7

d. 7

Describe the pattern, then find the next two numbers. 7, 12, 17, 22, ... a. add 5; 27, 35 b. add 4; 27, 40 c. add 6; 29, 32 d. add 5; 27, 32

d. add 5; 27, 32

Describe the pattern and give the next two numbers in the sequence. 7, 13, 19, 25,... a. add 7; 33, 37 b. add 6; 31, 40 c. add 5; 31, 46 d. add 6; 31, 37

d. add 6 31, 37

How do you find the common difference of a sequence?

d=aᴺ-aᴺ-¹

aⁿ

n-th Term of a Sequence

The sum of the first "n" terms of an infinite series is called a...

partial sum (of the series)

How do you find the common ratio of a sequence?

r=aᴺ/aᴺ-¹

When you are solving for sigma notation problems and they are in the form of an arithmetic sequence, then you can use the formula ... ?

sᴺ=n(a¹+aᴺ)/2 (you still have to find the first and last term before plugging it into the formula)

common difference

the difference between any two consecutive terms in an arithmetic sequence