Mathematical Patterns & Sequences

Suppose you drop a handball from a height of 10 feet. After the ball hits the floor, it rebounds to 85% of its previous height. How high will the ball rebound after its fourth bounce? After what bounce will the rebound height be less then 2 feet?

BouncesHeight  10 = 8.5 ft 2.85  8.5 =  =  = The ball will rebound about 5.2 feet on the fourth bounce. This is an example of a sequence…

Vocabulary Sequence – an ordered list of numbers. Term – each individual number of a sequence. We denote the terms of a sequence using a variable, such as a, with a positive integer subscript. a 1, a 2, a 3, a 4,…a n-1, a n, a n+1

Vocabulary Recursive Formula – a formula for the sequence which defines the terms in a sequence by relating each term to the terms before it. Explicit Formula – a formula that expresses the nth term in terms of n.

Describe the pattern: -14, -8, -2, 4, 10… Adding 6. Therefore, the recursive formula is: a n = a n-1 + 6, where a 1 = -14

We could use the recursive formula to predict the 8 th number in the sequence. -14, -8, -2, 4, 10… a n = a n a 8 = a since a 7 = a a 8 = (a 6 + 6) + 6 since a 6 = a a 8 = ((a 5 + 6) + 6) + 6 a 8 = ((10 + 6) + 6) + 6= 28

Describe the pattern: 2,6,18,54,162 … Multiplying by 3. Therefore, the recursive formula is: a n = 3a n-1, where a 1 = 2

We could use the recursive formula to predict the 10 th number in the sequence. 2,6,18,54,162… a n = 3a n-1 a 10 = 3a 9 since a 9 = 3a 8 a 10 = 3(3a 8 ) since a 8 = 3a 7 a 10 = 9a 8 = 9(3a 7 ) = 27a 7 since a 7 = 3a 6

2,6,18,54,162… a n = 3a n-1 a 10 = 27a 7 = 27(3a 6 ) = 81a 6 since a 6 = 3a 5 a 10 = 81a 6 =81(3a 5 ) = 243a 5 = 243(162)= 39,366 since a 7 = 3a 6

Complete pg. 591:

a1a2a3a4 Length of a Side 1234 Perimeter The table above shows the perimeters of regular pentagons with sides from 1 to 4 units long. The numbers in each row forms a sequence.

a1a2a3a4 Length of a Side 1234 Perimeter Let’s write an explicit formula for the perimeter sequence. Row 2: Think of them as ordered pairs. (1,5),(2,10),(3,15) and (4,20) Term 1Term 2Term 3Term 4

(1,5),(2,10),(3,15) and (4,20) Find the slope: m = y 2 – y 1 x 2 – x 1 m = 10 – 5 2 – 1 m = 5151 m = 5 Find the equation: y – y 1 = m(x – x 1 ) y – 5 = 5(x – 1) y – 5 = 5x – 5 y = 5x Write your explicit formula in terms of a n a n = 5n

Write and explicit formula for the following sequence. Think of them as ordered pairs. (1,17),(2,8),(3,-1) and (4,-10) Term 1Term 2Term 3Term 4 17, 8,-1,-10…

Find the slope: m = y 2 – y 1 x 2 – x 1 m = -1 – 8 3 – 2 m = -9 1 m = -9 Find the equation: y – y 1 = m(x – x 1 ) y – 8 = -9(x – 2) y – 8 = -9x + 18 y = -9x + 26 Write your explicit formula in terms of a n a n = -9n , 8,-1,-10… (1,17),(2,8),(3,-1) and (4,-10)

Complete pg. 591:

Decide whether the following formula is explicit or recursive? a 1 = -2; a n = a n List the first five numbers of the sequence.

Decide whether the following formula is explicit or recursive? a n = n List the first five numbers of the sequence.

Decide whether the following formula is explicit or recursive? a n = 3n(n+1) List the first five numbers of the sequence.

Decide whether the following formula is explicit or recursive? a 1 =121; a n = a n List the first five numbers of the sequence.

Assignment: pg. 591: 24-31, even,

Vocabulary Arithmetic Sequence – a sequence where the difference between two consecutive terms in constant. Common Difference – the difference between two consecutive terms of an Arithmetic Sequence. Example: 48, 45, 42, 39… -3 Common Difference is -3

Vocabulary Geometric Sequence – a sequence where the ratio between consecutive terms is constant. Common Ratio – the ratio between two consecutive terms of a Geometric Sequence. Example: 5, 15, 45, 135… x3 Common Ratio is 3

Are the following sequences Arithmetic, Geometric, or Neither? 15, 30, 45, 60… Geometric: CR = 3 37, 34, 31, 28, 25, … 700, 350, 175, 87.5,… 1, 1 / 2, 1 / 3, 1 / 4, 1 / 5,… Arithmetic; CD=15 2, 6, 18, 54, … Arithmetic; CD=-3 Geometric: CR = 1 / 2 Neither Answer

Vocabulary Arithmetic Mean– the average of two numbers. Geometric Mean– the positive square root of the product of the two numbers.

Examples: Determine the arithmetic mean between 10 and

Examples: Determine the geometric mean between 6 and 12.  6  12  72 62626262

Examples: Determine the missing number of the following arithmetic sequence ,, 21

Examples: Determine the missing number of the following geometric sequence. 12,, 3  12  3  36 6