Sequence an ordered list of numbers Finding Patterns

Sequencean ordered list of numbers

Finding Patterns ArithmeticGeometric

Arithmetic Sequence

Pattern adding a fixed number from one term to the next

COMMON DIFFERENCEd

NOTEn increases byonly 1 at a time n

an =

Examples

Sequence A: 5 , 8 , 11 , 14 , 17 , ... Sequence B: 26 , 31 , 36 , 41 , 46 , ... Sequence C: 20 , 18 , 16 , 14 , 12 , ...

Common Difference - the fixed numbers that binds the sequence together

In Sequence A the common difference is +3

In Sequence B the common difference is +5

In Sequence C the common difference is -2

Common Difference = d Generic sequence is referred to using the letter a along with subscripts as follows:

Generic Sequence: a1, a2, a3, a4, ...

The fifth term of a given sequence = a5.

The 17th term = a17

The nth term = an The term right before the nth term = an-1

d can be calculated by subtracting any two consecutive terms in an arithmetic sequence.

d = an - an - 1, where n is any positive integer greater than 1.

Common difference (d) = 3

an = dn + c or an = 3n + cc is some number that must be found

Common Difference = 5Formula for the nth term = an = 5n + 21

14th terma14 = 5(14) + 21 = 70 + 21 = 91

40th terma40 = 5(40) + 21 = 200 + 21 = 221

Common difference = -2. Formula will be -2n + cFind c

-2×1 + c = -2 + c-2×2 + c = -4 + c-2×3 + c = -6 + c-2×4 + c = -8 + c-2×5 + c = -10 + c

C = 22

Geometric Sequence

Pattern Multiply a fixed number from one term to the next

Common Ratio r

an=

n

NOTEn increases byonly 1 at a time

a1 a2 a3 a4

5, 10, 20, 40, ...multiply each term by 2 to arrive at the next term

or...divide a2 by a1 to find the common ratio, 2

r = 2

-11, 22, -44, 88, ...

multiply each term by -2 to arrive at the next term

or...divide a2 by a1 to find the common ratio, -2

r = -2

Find the common ratio for the sequence

r = divide the second term by the first term = -1/2.

Checking shows that multiplying each entry by -1/2 yields the next entry.

Find the 7th term of the sequence2, 6, 18, 54, ...

n = 7; a1 = 2, r = 3

The seventh term is 1458.

Find the 11th term of the sequence 1 1 1

1, , ,2 4 8

n = 11; a1 = 1, r = -1/2

A ball is dropped from a height of 8’. The ball bounces to 80% of its previous height with each bounce.

How high (to the nearest tenth of a foot) does the ball bounce on the fifth bounce?

Set up a model drawing for each "bounce". 6.4, 5.12, ___, ___, ___ The common ratio is 0.8.

Answer: 2.6 feet

Finding More Patterns Fractal

Carl Friedrich GaussGermany(1777 - 1855)

Add the integers from 1 to 100

There are 50 pairs of 101...

Add the integers from 1 to 100

Do you think we can find a formula that will work for adding all the integers from 1 to n?

How many pairs of n + 1 are there? Half of n!

5050 + 101 = 5151

Do you think we can find a formula that will work for adding all the integers from 1 to n?

Write this series in sigma notation?

A similar formula works for when the terms skip some numbers, like

To :

Find the sum of the first n terms

This is for arithmetic sequences ONLY!Let's find the sum of the first 50 terms of the arithmetic sequence

We need:

We have:

Finding More Patterns Fractal

Koch’s SnowflakeFractal Pattern

What is its perimeter?

Start with an equilateral triangle.To each side….

Koch’s SnowflakeFractal Pattern

Divide each side into thirds…The side length of each successive small triangle is 1/3

What is its perimeter?

Koch’s SnowflakeFractal

Total length increases by one third and thus the length at step n will be (4/3)n