11.1 Mathematical Patterns

Ex 1 Start with a square with sides 1 unit long. On the right side, add on a square of the same size. Continue adding one square at a time in this way. Draw the first four figures.

Ex 2 Write the number of 1 unit segments in each figure from ex 1 as a sequence.

Ex 3 Describe the pattern formed and find the next three terms. 243, 81, 27, 9, …

Ex 4 Suppose you drop a ball from a height of 100 cm. It bounces back to 80% of its previous height. How high will it go after its fifth bounce?

We can use a variable with positive integer subscripts to represent the terms in a sequence: a1 a2 a3 – first, second and third terms an-1: n – 1 term an : nth term an+1 : n + 1 term n is the term number

Recursive formula Defines the terms in a sequence by relating each term to the ones before it. (ex 4 was recursive b/c the height was 80% of its previous height) Formula would be an = 0.80an-1 where a1 = 100

Ex 5 Describe the pattern of the sequence: 2, 6, 18, 54, 162, … Write a recursive function. Find the 6th and 7th terms. Find the value of a10

Explicit formula Expresses the nth term in terms of n Finding the value of a term without knowing the preceding term. (Find a link between the term number and the term value.)

EX 6 Write a formula. 2, 6, 12, 20, …

Ex 7 Write a formula 3, 5, 7, 9, …

Ex 8 Term a1 a2 a3 a4 Length of side 1 2 3 4 perimeter 5 10 15 20 For each sequence, find the next term and the 20th term. Write an explicit formula for each sequence. n = the term number

Ex 9 Write the first six terms of the area of squares that have side lengths 1, 2, 3, etc. Write an explicit formula.

Ex 10 Write a formula for

Ex 11 Write a formula for: