MDME: MANUFACTURING, DESIGN, MECHANICAL ENGINEERING

# 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-and-series-01.pdf

## SEQUENCES

#### Arithmetic Sequence

Notation

xn is the nth term in a sequence

For example
x1 is the 1st term in a sequence
x6 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:

xn = a + (n-1)d

Where:

• xn is the nth term in the sequence
• a is the first term (x1)
• 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. xn = a + (n-1)d

Here, a = 4 and d = 7.
So xn = 4 + 7(n-1)
Therefore the 26th term would be:
x26 = 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. xn = a + (n-1)d
It this case: xn = 12 + -9(n – 1) = 12 – 9(n – 1)
Therefore the 13th term would be:
x13 = 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, ar2, ar3, . . . }

A rule for geometric sequences

xn = a r(n-1)

Where:
xn 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. xn = ar(n-1)
Here, a = 2, r = 3
So: xn = 2 x 3(n-1)
Therefore the 12th term would be:
x26 = 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. xn = ar(n-1)
Here, a = 128, r = 0.5
So: xn = 128 x 0.5(n-1)
Therefore the 10th term would be:
x26 = 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
a1 = first term
an = the nth term

Example:

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

n = 5
a1 = 2
a5 = 14

So S5 = 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

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

S5 = 2(1-54) / (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 mth term to the nth 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