Contents

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Arithmetic and Geometric Sequences
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Contents
1.1 Sequences
1.2 Arithmetic Sequence and Geometric
Sequence
1.3 Summing a Sequence
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1.1 Sequences
A sequence is a number pattern in a definite
order. Each number in a sequence is called a
term. For example, in the sequence of triangular
numbers, 1, 3, 6, 10, 15,,
The first term is 1, which is usually denoted by
T(1). Similarly, the other terms can be denoted
by T(2), T(3), etc. the () at the end means
that the sequence continues infinitely.
In the sequence of triangular numbers, the nth is
given by the formula
This is called the general term of the sequence
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1.1 Sequences
This sequence with a common ratio between
consecutive terms is called a geometric sequence
( or geometric progression, G.P.)
Apart from geometric sequence, there is another
well-know sequence. This kind of sequence have a
common difference between consecutive terms, and
is called arithmetic sequence ( or arithmetic
progression, A.P).
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1.2 Arithmetic Sequence and Geometric Sequence
A. Arithmetic Sequence
Consider the sequence 1, 4, 7, 11, 15,. In this
sequence, every term after the first term is
larger than its preceding term by 3, therefore,
the difference 3 is called the common difference
of the sequence and this kind of sequence is
called an arithmetic sequence.
Let the common difference be d and the first term
be a. We have
T(1) a, T(2) a d, T(3) a 2d, , T(n)
a (n 1)d, .
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1.2 Arithmetic Sequence and Geometric Sequence
B. Geometric Sequence
If each term of a sequence differs from the
preceding term by the same multiplying factor,
this is called a geometric sequence. Moreover,
the same multiplying factor is called the common
ratio, usually denoted by r.
If the first term of a geometric sequence is
denoted by a, we have
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1.3 Summing a Sequence
The sum of the terms of a sequence is called a
series. Consider the sequence T(1), T(2), T(3),
, T(n)
Let S(n) T(1) T(2) T(3) T(n). S(n) is
called a series of n terms
In other words, S(n) is the sum of the first n
terms of sequence. For example, in the sequence
specified by the general term T(n) (1)n n2,
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1.3 Summing a Sequence
A. Summing an Arithmetic Sequence
When we do not know the last term l, we may
substitute l a (n 1)d into the formula of
the sum of an arithmetic series.
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1.3 Summing a Sequence
B. Summing a Geometric Sequence
For a geometric series whose first term is a and
the common ratio is r, the sum to n terms is
given by
This formula works best when r gt 1 so that the
denominator will not be negative.
If r lt 1, we may use the equivalent formula
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1.3 Summing a Sequence
B. Summing to infinity
A concept which makes mathematics so fascinating
is the concept of infinity. infinite sum is one
of the problems explored through generations.