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 ^3^3

3, 10, 29, 66, 127, 218 ^7^7 ^ 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, 218 ^7^7 ^ 19 ^ 37 ^ 61 ^ 91 ^ 18 ^ 24 ^ 30 ^ 12 ^6^6 ^6^6 ^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 sequence 1.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, 14 Difference: 2 Add: 4 nth term: 2n + 4

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

Part 2

Finding the nth term of a quadratic sequence 1.List the factors of each term in the sequence 2.Look for a linear pattern that uses one factor from each term 3.Look for a linear pattern in the other factor from each term 4.Write the formula for the nth term for each 5.Multiply them together Note that ANY 2 nd degree polynomial can be written in the Quad. Equation, y = ax 2 + bx + c ANY quadratic will have this form, (remember in the Quad Formula, ax 2 + bx + c = 0 when using a, b, & c.) though sometimes b or c is 0. examples: 5x 2 – 2x + 3, x 2 – 4x, -3x What ARE a, b, & c in each of these quadratic expressions?

1.List the factors of each term in the sequence 2.Look for a linear pattern that uses one factor from each term 3.Look for a linear pattern in the other factor from each term 4.Write the formula for the nth term for each 5.Multiply them together Our problems start with a chart showing the sequence f(n) for each value of n. (remember in the Quad Formula, ax 2 + bx + c = 0 when using a, b, & c.) So, let’s change the quadratic equation to match our problem: y = ax 2 + bx + c input output f(n) = an 2 + bn + c changes to or an 2 + bn + c = f(n) and we need to find the nth term

1.List the factors of each term in the sequence 2.Look for a linear pattern that uses one factor from each term 3.Look for a linear pattern in the other factor from each term 4.Write the formula for the nth term for each 5.Multiply them together (remember in the Quad Formula, ax 2 + bx + c = 0 when using a, b, & c.) input output A) Write 3 equations by replacing n in the above equation with 1, then 2, then 3. STEP Example. #1 a(1 2 ) + b(1) + c = 6 #2 a(2 2 ) + b(2) + c = 19 #3 a(3 2 ) + b(3) + c = 42 B) Simplify each equation #1 a + b + c = 6 #2 4a + 2b + c = 19 #3 9a + 3b + c = 42 an 2 + bn + c = f(n) Set each equation equal the f(n) from the chart.

1.List the factors of each term in the sequence 2.Look for a linear pattern that uses one factor from each term 3.Look for a linear pattern in the other factor from each term 4.Write the formula for the nth term for each 5.Multiply them together (remember in the Quad Formula, ax 2 + bx + c = 0 when using a, b, & c.) input output 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 #1 a + b + c = 6 #2 4a + 2b + c = 19 #3 9a + 3b + c = 42 Example: and subtract equation #2 – #1,

1.List the factors of each term in the sequence 2.Look for a linear pattern that uses one factor from each term 3.Look for a linear pattern in the other factor from each term 4.Write the formula for the nth term for each 5.Multiply them together (remember in the Quad Formula, ax 2 + bx + c = 0 when using a, b, & c.) input output E) Using the equation 3a + b = 13 3(5) + b = 13 substitute 5 in for a b = -2 STEP example: #1 a(1 2 ) + b(1) + c = 6 #2 a(2 2 ) + b(2) + c = 19 #3 a(3 2 ) + b(3) + c = 42 G) Write your final equation using a, b, and c into the quadratic equation. (Check your answer.) a = 5, b = -2, c = 3 so the nth term is: 5n 2 – 2n + 3 Find a  a = 5 F) Now sub a & b into equation #1 #1  a + b + c = 6 and find c c = 6 c = 3 2a = 10

Practice: 8, 15, 24, 35, 48, 63, 80 Put in the n’s to go with this sequence (click for answer) 

nth term for the quadratic: n 2 + 4n + 3

Example: -8, 11, 42, 85, 140, 207, 286 (click for answer) 

nth term for the quadratic: 6n 2 + 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

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.