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

3, 10, 29, 66, 127, 2187̂

^19

^37

^61

^91

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.

3, 10, 29, 66, 127, 2187̂

^19

^37

^61

^91

^18

^24

^30

^12

^6

^6

^6

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.

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)

Example

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

Example

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

Part 2

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?

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

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.

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,

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

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)

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

Example:

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

(click for answer)

nth term for the quadratic:

6n2 + n – 15

END QUADRATICS

Finding the nth term of a cubic (or quartic)

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

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.

Finding the nth term of a geometric sequence

Use this formula!

• = first term• r = common ratio

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.