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
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
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
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