Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between...

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Sequence s and Series!!!

Transcript of Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between...

Page 1: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Sequences and

Series!!!

Page 2: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Finding the Degree of a Sequence

Begin by finding the difference between adjacent numbers

Page 3: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

2, 5, 8, 11, 14^3

^3

^3

^3

Page 4: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

3, 10, 29, 66, 127, 2187̂

^19

^37

^61

^91

Page 5: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

If the number you add each time is the same, you are done. YAY!!!

If the number you add each time is different, repeat the process- find the difference between the differences. Continue until they are the same.

Page 6: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

3, 10, 29, 66, 127, 2187̂

^19

^37

^61

^91

^18

^24

^30

^12

^6

^6

^6

Page 7: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

If you complete this step….

•One time, the sequence is linear.• Two times, the sequence is quadratic. • Three times, the sequence is cubic.• Four times, the sequences is quartic.

Page 8: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Finding the nth term of a linear sequence1. Find the difference between the numbers. 2. Find out what you would have to add or subtract to

your difference, in order to get your start value. 3. Write your formula:

(difference)*n + (the number you add/subtract)

Page 9: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Example

6, 8, 10, 12, 14Difference: 2Add: 4nth term: 2n + 4

Page 10: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Example

4, 1, -2, -5, -8Difference: -3Add: 7nth term: -3n + 7

Page 11: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Part 2

Page 12: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Finding the nth term of a quadratic sequence

Note that ANY 2nd degree polynomial can be written in the Quad. Equation, y = ax2 + bx + c

ANY quadratic will have this form, though sometimes b or c is 0.

examples: 5x2 – 2x + 3, x2 – 4x, -3x2 + 7

What ARE a, b, & c in each of these quadratic expressions?

Page 13: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Our problems start with a chart showing the sequence f(n) for each value of n.

So, let’s change the quadratic equation to match our problem: y = ax2 + bx + c

n 1 2 3 4 5f(n) 6 19 42 75 118

inputoutput

f(n) = an2 + bn + cchanges to

or an2 + bn + c = f(n)

and we need to find the nth term

Page 14: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

n 1 2 3 4 5f(n) 6 19 42 75 118

inputoutput

A) Write 3 equations by replacing n in the above equation with 1, then 2, then 3.

STEP Example .

#1 a(12) + b(1) + c = 6

#2 a(22) + b(2) + c = 19

#3 a(32) + b(3) + c = 42

B) Simplify each equation

#1 a + b + c = 6

#2 4a + 2b + c = 19

#3 9a + 3b + c = 42

an2 + bn + c = f(n)

Set each equation equal the f(n) from the chart.

Page 15: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

n 1 2 3 4 5f(n) 6 19 42 75 118

inputoutput

STEP

C) Subtract equation #3 – #2

4a + 2b + c = 19 – ( a + b + c = 6)

3a + b = 13

D) Then subtract those two answers from each other.

5a + b = 23– (3a + b = 13)

2a = 10

9a + 3b + c = 42 – (4a + 2b + c = 19)

5a + b = 23

#1 a + b + c = 6

#2 4a + 2b + c = 19

#3 9a + 3b + c = 42 Example:

and

subtract equation #2 – #1,

Page 16: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

n 1 2 3 4 5f(n) 6 19 42 75 118

inputoutput

E) Using the equation 3a + b = 13 3(5) + b = 13 substitute 5 in for a b = -2

STEP example:

G) Write your final equation using a, b, and c into the quadratic equation. (Check your answer.)

a = 5, b = -2, c = 3so the nth term is:

5n2 – 2n + 3

Find a a = 5

F) Now sub a & b into equation #1 #1 a + b + c = 6 and find c 5 + -2 + c = 6 c = 3

2a = 10

Page 17: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Practice:

8, 15, 24, 35, 48, 63, 80

Put in the n’s to go with this sequence

1 2 3 4 5 6 7

(click for answer)

Page 18: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

nth term for the quadratic:n2 + 4n + 3

Page 19: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Example:

-8, 11, 42, 85, 140, 207, 286

(click for answer)

Page 20: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

nth term for the quadratic:

6n2 + n – 15

Page 21: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

END QUADRATICS

Page 22: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Finding the nth term of a cubic (or quartic)

Use more equations to find a b c & d!!!

Page 23: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Arithmetic vs. Geometric Sequences

Arithmetic is just another name for a linear sequence.

Geometric is where the ratio (what you multiply) between consecutive terms is constant. The number you multiply by is called the common ratio.

Page 24: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Finding the nth term of a geometric sequence

Use this formula!

• = first term• r = common ratio

Page 25: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.

Series vs. Sequences

A sequence is just a list of numbers that follow a pattern. A series is like a sequence, but the numbers are added together.

Page 26: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.
Page 27: Sequences and Series!! !. Finding the Degree of a Sequence Begin by finding the difference between adjacent numbers.