Marco's Problem

A Sequence is a list of things (usually numbers) that are in order.

Infinite or Finite When the sequence goes on forever it is called an infinite sequence,otherwise it is a finite sequence. Examples: {1, 2, 3, 4, ...} is a very simple sequence (and it is an infinite sequence)

{20, 25, 30, 35, ...} is also an infinite sequence

{1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite sequence)

{4, 3, 2, 1} is 4 to 1 backwards

{1, 2, 4, 8, 16, 32, ...} is an infinite sequence where every term doubles

{a, b, c, d, e} is the sequence of the first 5 letters alphabetically

{f, r, e, d} is the sequence of letters in the name "fred"

{0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s (yes they are in order, it is an alternating order in this case)

In Order When we say the terms are "in order", we are free to define what order that is! They could go forwards, backwards ... or they could alternate ... or any type of order we want!

Like a Set A Sequence is like a Set, except:

the terms are in order (with Sets the order does not matter) the same value can appear many times (only once in Sets)

Example: {0, 1, 0, 1, 0, 1, ...} is the sequence of alternating 0s and 1s. The set is just {0,1}

Notation To make it easier to use rules, we often use this special style:

Example: to mention the "5th term" we write: x5 So a rule for {3, 5, 7, 9, ...} can be written as an equation like this:

xn = 2n+1

And to calculate the 10th term we can write:

x10 = 2n+1 = 2×10+1 = 21

Can you calculate x50 (the 50th term) doing this?

Here is another example:

Example: Calculate the first 4 terms of this sequence:

{an} = { (-1/n)n }

Calculations:

a1 = (-1/1)1 = -1 a2 = (-1/2)2 = 1/4 a3 = (-1/3)3 = -1/27 a4 = (-1/4)4 = 1/256 Answer:

{an} = { -1, 1/4, -1/27, 1/256, ... }