# SEQUENCES & SERIES

A **sequence** is a list of numbers. An arithmetic sequence of numbers changes in a straight line, and a geometric sequence changes in a curve. A **series** is the sum of a sequence.

Smartboard Notes:

## SEQUENCES

#### Arithmetic Sequence

**Notation**

**x _{n}** is the

**n**th term in a sequence

*For example
x _{1} is the 1st term in a sequence
x_{6} is the 6th term in a sequence *

**Defining an arithmetic sequence **

An arithmetic sequence is formed when the gap between each term is the same (or a constant).

An arithmetic sequence could be shown like this:

{ a, a+d, a+2d, a+3d, . . . }

To find the value of any term:

**x _{n} = a + (n-1)d**

Where:

**x**is the nth term in the sequence_{n}

**a**is the first term (x_{1})**d**is the common difference**n**is the number of the term

**Example 1**

Consider the sequence

{ 4, 11, 18, 25, 32, 39, . . . }

And you want to find the 26th term

Write a rule to express the term i.e. x_{n} = a + (n-1)d

Here, a = 4 and d = 7.

So x_{n} = 4 + 7(n-1)

Therefore the 26th term would be:

x_{26} = 4 + 7(26-1) = 179

**Example 2**

Consider the sequence

{ 12, 3, -6, -15, . . . }

And you want to find the 13th term

Write a rule to express the term i.e. x_{n} = a + (n-1)d

It this case: x_{n} = 12 + -9(n – 1) = 12 – 9(n – 1)

Therefore the 13th term would be:

x_{13} = 12 – 9(13 – 1) = -96

#### Geometric Sequence

Defining a geometric sequence

In a geometric sequence, each term is found by multiplying the previous term by a constant.

The multiplying constant is called the common ratio

A geometric sequence could be shown like this:

{ a, a*r, (a*r)*r, (a*r*r)*r, . . . } or

{ a, ar, ar^{2}, ar^{3}, . . . }

A rule for geometric sequences

x_{n} = a r^{(n-1)}

*Where:
x_{n} is the nth term in the sequence
a is the first term
n is the number of the term
r is the common ratio*

**Example 1**

Consider the sequence

{ 2, 6, 18, 54, 162, 486, . . . }

And you want to find the 12th term

Write a rule to express the term i.e. x_{n} = ar^{(n-1)}

Here, a = 2, r = 3

So: x_{n} = 2 x 3^{(n-1)}

Therefore the 12th term would be:

x_{26} = 2 x 3^{(12-1)} = 354294

**Example 2**

Consider the sequence

{ 128, 64, 32, 16, 8, . . . }

And you want to find the 12th term

Write a rule to express the term i.e. x_{n} = ar^{(n-1)}

Here, a = 128, r = 0.5

So: x_{n} = 128 x 0.5^{(n-1)}

Therefore the 10th term would be:

x_{26} = 128 x 0.5^{(10-1)} = 0.25

## SERIES

####
Arithmetic Series

The sum of the members of a finite arithmetic progression is called an **arithmetic series**.

The formula is;

*Where;*

**n** = number of terms

**a _{1}** = first term

**a**= the nth term_{n}

**Example:**

Find this sum: 2 + 5 + 8 + 11 + 14

**n** = 5

**a _{1}** = 2

**a**= 14_{5}So S_{5} = 5/2 * (2 + 14) = 404

Here are some useful shortcuts for the summing a standard sequence.

The three formulas represent a series like this:

1+2+3+4+5... (arithmetic)

1+4+9+16+25... (not geometric but a power of 2 sequence)

1+8+27+64+125... (not geometric but a power of 3 sequence)

The main one we want here is the arithmetic finite sum where d=1 and a=1;

If we want to find the sum from m to n, we simply do the sum from 1 to n and substract the sum from 1 to m

**Example:**

Find the partial sum of the series of numbers from 10 to 90

Substitute from the formula for {k}

#### Geometric Series

##### Finite Geometric Series

A **geometric series** is the sum of the numbers in a finite geometric progression starting from 1.

Note: This is read as: "The sum from 1 to n of ar^{(k-1)} is..."

*Where;*

**n** = number of terms

**a** = first term

**r** = the common ratio

**k** = the summing counter

**Example:**

2 + 10 + 50 + 250

**n** = 4

**a** = 2

**r** = 5

*S _{5} = 2(1-5^{4}) / (1-5)* = -1248 / -4 = 312

If we want the progression to start from a number other than 1, say m, then we sum from 1 and subtract the sum to m;

So the sum of the sequence from the *m*th term to the *n*th term is;

^{}

##### Infinite Geometric Series

An **infinite geometric series** is an infinite series whose successive terms have a common ratio. Such a series only converges if the absolute value of the common ratio is less than one (|*r*| < 1).

Example: A geometic series where a = 0.5, and r = 0.5 converges to 1;

(This converges because r is less than 1)

The convergence value of the sum of an infinite geometric series can be computed from;

Example: The same geometic series (where a = 0.5, and r = 0.5);

= 0.5 / (1-0.5) = 0.5/0.5 = 1

**Why r must be less than 1**

Of course, if r>1 then the sum of an infinite series will be infinity.

1+2+4+8+16+....to infinity = infinity

#### Questions:

Assignment: Do all questions