Topic 1 Arithmetic Sequences And Series

What about guess the rule? Look at these number sequences carefully can you guess the next 2 numbers? What about guess the rule? +10 30 40 50 60 70 80 +3 --------------------------------------------------------------------------------------------------------------------- 17 20 23 26 29 32 --------------------------------------------------------------------------------------------------------------------- -7 48 41 34 27 20 13

Can you work out the missing numbers in each of these sequences? +25 50 75 100 125 150 175 --------------------------------------------------------------------------------------------------------------------- +20 30 50 70 90 110 130 --------------------------------------------------------------------------------------------------------------------- -5 196 191 181 176 171 186 --------------------------------------------------------------------------------------------------------------------- -10 306 296 286 276 266 256

Now try these sequences – think carefully and guess the last number! +1, +2, +3 … 1 2 4 7 11 16 --------------------------------------------------------------------------------------------------------------------- double 3 6 12 24 96 48 --------------------------------------------------------------------------------------------------------------------- + 1.5 0.5 2 3.5 5 6.5 8 --------------------------------------------------------------------------------------------------------------------- -3 7 4 1 -2 -5 -8

This is a really famous number sequence which was discovered by an Italian mathematician a long time ago. It is called the Fibonacci sequence and can be seen in many natural things like pine cones and sunflowers!!! 0 1 1 2 3 5 8 13 21 etc… Can you see how it is made? What will the next number be? 34!

For these sequences I have done 2 math functions! Guess my rule! For these sequences I have done 2 math functions! 3 7 15 31 63 127 2x + 1 2x - 1 2 3 5 9 17 33

What is a Number Sequence? A list of numbers where there is a pattern is called a number sequence The numbers in the sequence are said to be its members or its terms.

To write the terms of a sequence given the nth term Sequences To write the terms of a sequence given the nth term Given the expression: 2n + 3, write the first 5 terms In this expression the letter n represents the term number. So, if we substitute the term number for the letter n we will find value that particular term. The first 5 terms of the sequence will be using values for n of: 1, 2, 3, 4 and 5 term 5 term 4 term 1 term 3 term 2 2 x 5 + 3 2 x 1 + 3 2 x 3 + 3 2 x 2 + 3 2 x 4 + 3 13 5 9 11 7

Write the first 3 terms of these sequences: 1) n + 2 2) 2n + 5 Now try these: Write the first 3 terms of these sequences: 1) n + 2 2) 2n + 5 3) 3n - 2 4) 5n + 3 5) -4n + 10 6) n2 + 2 3, 4, 5 7, 9, 11 1, 4, 7 8, 13, 18 6, 2, - 2, 3, 6, 11,

6B - The General Term of A Number Sequence Sequences may be defined in one of the following ways: listing the first few terms and assuming the pattern represented continues indefinitely giving a description in words using a formula which represents the general term or nth term.

The first row has three bricks, the second row has four bricks, and the third row has five bricks. If un represents the number of bricks in row n (from the top) then u1 = 3, u2 = 4, u3 = 5, u4 = 6, ....

This sequence can be describe in one of four ways: Listing the terms: u1 = 3, u2 = 4, u3 = 5, u4 = 6, ....

This sequence can be describe in one of four ways: Using Words: The first row has three bricks and each successive row under the row has one more brick...

This sequence can be describe in one of four ways: Using an explicit formula: un = n + 2 u1 = 1 + 2 = 3 u2 = 2 + 2 = 4 u3 = 3 + 2 = 5 u4 = 4 + 2 = 6, ....

This sequence can be describe in one of four ways: Using a graph

What you really need to know! An arithmetic sequence is a sequence in which the difference between any two consecutive terms, called the common difference, is the same. In the sequence 2, 9, 16, 23, 30, . . . , the common difference is 7.

An Arithmetic Sequence is defined as a sequence in which there is a common difference between consecutive terms.

What you really need to know! A geometric sequence is a sequence in which the quotient of any two consecutive terms, called the common ratio, is the same. In the sequence 1, 4, 16, 64, 256, . . , the common ratio is 4.

Example 1: State whether the sequence -5, -1, 3, 7, 11, … is arithmetic. If it is, state the common difference and write the next three terms.

Arithmetic! + 4 11 – 7 4 7 – 3 3 – -1 -1 – -5 -5, -1, 3, 7, 11, Example 2: -5, -1, 3, 7, 11, 15, 19, 23 Subtract Common difference 11 – 7 4 7 – 3 3 – -1 -1 – -5 Arithmetic! + 4

Example 2: State whether the sequence 0, 2, 6, 12, 20, … is arithmetic. If it is, state the common difference and write the next three terms.

Example 2: 0, 2, 6, 12, 20 … Subtract Common difference 20 – 12 8 12 – 6 6 6 – 2 4 2 – 0 2 Not Arithmetic!

Example 3: State whether the sequence 2, 4, 4, 8, 8, 16, 16 … is geometric. If it is, state the common ratio and write the next three terms.

Example 3: 2, 4, 4, 8, 8, 16, 16, … Divide Common ratio 16 ÷ 16 1 16 ÷ 8 2 8 ÷ 8 8 ÷ 4 4 ÷ 4 4 ÷ 2 Not Geometric!

Example 4: State whether the sequence 27, -9, 3, -1, 1/3, … is geometric. If it is, state the common ratio and write the next three terms.

1/3 ÷ -1 -1/3 -1 ÷ 3 3 ÷ -9 -9 ÷ 27 Geometric! • -1/3 Example 4: 27, -9, 3, -1, 1/3, -1/9, 1/27, -1/81 Divide Common ratio 1/3 ÷ -1 -1/3 -1 ÷ 3 3 ÷ -9 -9 ÷ 27 Geometric! • -1/3

Which of the following sequences are arithmetic Which of the following sequences are arithmetic? Identify the common difference. YES YES NO NO YES

The common difference is always the difference between any term and the term that proceeds that term. Common Difference = 5

The general form of an ARITHMETIC sequence. First Term: Second Term: Third Term: Fourth Term: Fifth Term: nth Term:

Formula for the nth term of an ARITHMETIC sequence. If we know any three of these we ought to be able to find the fourth.

Given: Find: IDENTIFY SOLVE

If it’s not an integer, it’s not a term in the sequence Given: Find: What term number is (-169)? IDENTIFY SOLVE If it’s not an integer, it’s not a term in the sequence

Homework Page 156 2 - 11 ( Any 8 Problems) Take Home Test Due Tuesday.