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• Arithmetic sequences and series;
• Sum of finite arithmetic series;
• Geometric sequences and series;
• Sum of finite and infinite geometric series.
• Sigma notation.
• Applications
In mathematics it is important that we can:
• Recognize a pattern in a set of numbers,
• Describe the pattern in words, and
• Continue the pattern.
A list of numbers where there is a pattern is called a number sequence. The numbers in the sequence are said
to be its members or its terms.
THE GENERAL TERM OF A NUMBER SEQUENCE
Sequences may be defined in one of the following ways:
•
Listing the first few terms and assuming that the pattern represented continues indefinitely
•
Giving a description in words
•
Using an explicit formula that represents the general term or nth term.
•
Using a pictorial or graphical representation
THE GENERAL TERM
The general term or nth term of a sequence is represented by a symbol with a subscript, for example un, Tn, tn,
or An. The general term is defined for n = 1, 2, 3, 4, 5, 6, ....
{un} represents the sequence that can be generated by using un as the nth term. The general term un is a
function where n –› un, and the domain is n ! Z+.
ARITHMETIC SEQUENCES
An arithmetic sequence is a sequence in which each term differs from the previous one by the same fixed
number. It can also be referred to as an arithmetic progression.
ALGEBRAIC DEFINITION
{un} is arithmetic
, un +1 – un = d for all positive integers n where d is a constant called the common
difference.
The symbol
, is read as ‘if and only if’. It means that:
• if {un} is arithmetic then un +1 – un is a constant
• if un +1 – un is a constant then {un} is arithmetic.
THE NAME ‘ARITHMETIC’
If a, b and c are any consecutive terms of an arithmetic sequence,
then b – a = c – b {equating common differences}
therefore 2b = a + c and b = a + c
2
So, the middle term is the arithmetic mean of the terms on either side of it.
THE GENERAL TERM FORMULA
Suppose the first term of an arithmetic sequence is u1 and the common difference is d.
For an arithmetic sequence with first term u1 and common difference d the general term or nth term is
un =u1 +(n – 1)d.
GEOMETRIC SEQUENCES
A sequence is geometric if each term can be obtained from the previous one by multiplying by the same nonzero constant. A geometric sequence can also be referred to as a geometric progression.
ALGEBRAIC DEFINITION
for all positive integers n where r is a constant called the common
ratio.
THE NAME ‘GEOMETRIC’
THE GENERAL TERM FORMULA
Suppose the first term of a geometric sequence is u1 and the common ratio is r. Then u2 = u1 r, u3 = u2 r2, u4 =
u3 r3, and so on.
For a geometric sequence with first term u1 and common ratio r, the general term or nth term is un = u1 r n – 1.
GEOMETRIC SEQUENCE PROBLEMS
Problems of growth and decay involve repeated multiplications by a constant number. We can therefore
model the situations using geometric sequences.
In these problems we will often obtain an equation, which we need to solve for n. We can do this using the
equation solver on our calculator.
COMPOUND INTEREST
THE COMPOUND INTEREST FORMULA
SERIES
A series is the addition of the terms of a sequence. For the sequence {un} the corresponding series is u1 + u2 +
u3 + …. The sum of a series is the result when we perform the addition.
Given a series which includes the first n terms of a sequence, its sum is Sn = u1 + u2 + u3 + … + un.
SIGMA NOTATION
PROPERTIES OF SIGMA NOTATION
ARITHMETIC SERIES
An arithmetic series is the addition of successive terms of an arithmetic sequence.
SUM OF AN ARITHMETIC SERIES
If the first term is u1 and the common difference is d, the terms are u1, u1 + d, u1 + 2d, u1 + 3d, and so on. The
sum of an arithmetic series with n terms is
SUM OF A FINITE ARITHMETIC SERIES
If the first term is u1 and the common difference is d, the terms are u1, u1 + d, u1 + 2d, u1 + 3d, and so on.
GEOMETRIC SERIES
A geometric series is the addition of successive terms of a geometric sequence.
If we are adding the first n terms of a geometric sequence, we say we have a finite geometric series. If we are
adding all of the terms in a geometric sequence, which goes on and on forever, we say we have an infinite
geometric series.
SUM OF A FINITE GEOMETRIC SERIES
If the first term is u1 and the common ratio is r, then the terms are: u1, u1r, u1r2, u1r3, ....
For a finite geometric series with r ≠ 1,
SUM OF AN INFINITE GEOMETRIC SERIES
To examine the sum of all the terms of an infinite geometric sequence, we need to consider
when n gets very large. If /r/ > 1, the series is said to be divergent and the sum becomes infinitely large.
If /r/ < 1, or in other words -1< r <1, then as n becomes very large, rn approaches 0.
This means that Sn will get closer and closer to
We call this the limiting sum of the series. This result can be used to find the value of recurring decimals.