SEQUENCES & SERIES

Sequences, Series, Sigma Notation, Limits, FTC Introduction DIFFERENCE BETWEEN SEQUENCE AND SERIES This is a sequence:This is a series: A sequence is just a list of numbers A series is a list of numbers being added together They can be as long or short as you want, some can even go on forever Each number listed is called a TERM The first term, the second term etc. We say n th term when referring to terms in general Menu

Sequences, Series, Sigma Notation, Limits, FTC Introduction 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 … The first term is 1 The second term is 1 The third term is 2 The seventh term is13 Here is a sequence: NOTATION Menu Some books use “t” for term, others use “a”.

Sequences, Series, Sigma Notation, Limits, FTC Introduction A Sequence is just a list of numbers in a particular order A typical sequence question looks like this: Find the next 2 numbers in these sequences Menu

Sequences, Series, Sigma Notation, Limits, FTC Introduction While finding the next number in a pattern is easy for many sequences, some are not so obvious: How to identify the pattern and find the next number in a sequence: Menu What do you add to 2 to get to 6? Sometimes this makes it easier to see a pattern, if not, do it again! Then work backwards to get the next number

Sequences, Series, Sigma Notation, Limits, FTC Introduction Most of these problems are easier: Menu All the problems we do will be… + a number (arithmetic) OR x a number (geometric)

Sequences, Series, Sigma Notation, Limits, FTC Introduction Can you figure this out without dividing? Menu

You should now be able to do PAGE #2 part 1 in the yellow booklet:

Sequences, Series, Sigma Notation, Limits, FTC basics Every sequence can be described with an equation: Menu Plug numbers in for “n” to find the terms of the sequence. To find the first 5 terms…plug in 1 – 5 If you wanted to know the 27 th term, plug in 27.

Sequences, Series, Sigma Notation, Limits, FTC basics Find the first 5 terms of this sequence: Menu

Sequences, Series, Sigma Notation, Limits, FTC basics Find the first 4 terms of this sequence: Menu

(You can cheat by using the table feature in your calculator)

Sequences, Series, Sigma Notation, Limits, FTC basics Finding the pattern for basic sequences: Menu Find the next term in each of the following sequences:

You should now be able to do PAGE#2, part 2 in the yellow booklet.

Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu SEQUENCES A Sequence is a list of numbers in a particular order. We are going to look at 2 types of sequences: Arithmetic and Geometric

A sequence is ARITHMETIC if: Each term goes up/down by the same amount The number it goes up/down by Is called the common difference Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu

A sequence is GEOMETRIC if: Each term is multiplied/divided by the same amount The number it is multiplied by is called the common ratio Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu

You should now be able to do PAGE#3 in the yellow booklet.

FORMULAS FOR THE n th TERM ARITHMETIC: 1 st term n th term The common difference Put 4 in: Out comes 11, the 4 th term! Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu

FORMULAS FOR THE n th TERM ARITHMETIC: Put 4 in: Out comes 26, the 4 th term! Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu 1 st term n th term The common difference

FORMULAS FOR THE n th TERM ARITHMETIC: Put 4 in: Out comes 21, the 4 th term! Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu 1 st term n th term The common difference

FORMULAS FOR THE n th TERM GEOMETRIC: 1 st termn th term The common ratio Put 4 in: Out comes 54, the 4 th term! Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu

FORMULAS FOR THE n th TERM GEOMETRIC: Put 4 in: Out comes 40, the 4 th term! Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu 1 st termn th term The common ratio

FORMULAS FOR THE n th TERM GEOMETRIC: Put 4 in: Out comes 1, the 4 th term! Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu 1 st termn th term The common ratio

FORMULAS FOR THE n th TERM Sequences, Series, Sigma Notation, Limits, FTC Sequences Menu These are called EXPLICIT definitions. Next, we will talk about other ways to describe a sequence. (RECURSIVE definitions)

You should now be able to do PAGE#4 in the yellow booklet.