Finite Sequences and Series

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Unit 7

• Finite Sequences and Series

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7.1 Arithmetic and Geometric Sequences
A sequence of numbers, T1, T2, T3,is called an
arithmetic sequence if and only if Tn1 - Tn d,
for all n 1, 2, , where d is a constant (i.e.
independent of n) called the common difference.
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7.1 Arithmetic and Geometric Sequences
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7.1 Arithmetic and Geometric Sequences
Some properties of an arithmetic sequence
Let a be the first term, d be the common
difference, Tn be the nth term and Sn be
the sum of the first n terms
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7.1 Arithmetic and Geometric Sequences
Some properties of an arithmetic sequence
(3) a, b, c are in arithmetic sequence if and
only if , and b is called the
arithmetic mean of a and c. (4) If a, b, c, d,
are in arithmetic sequence, then a k, b
k, c k, d k, and a k, b k, c
k, d k,, are also in arithmetic
sequences with the same common difference
as that of the original one.
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7.1 Arithmetic and Geometric Sequences
Some properties of an arithmetic sequence
(5) If a, b, c, d, are in arithmetic
sequence, then ak, bk, ck, dk, and are
also in arithmetic sequences with a new
common difference.
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7.1 Arithmetic and Geometric Sequences
A sequence of non-zero numbers T1, T2, T3,is
called a geometric sequence if and only if
, for all n 1, 2, .,where r is a constant
(i.e. independent of n) called the common ratio.
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7.1 Arithmetic and Geometric Sequences
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7.1 Arithmetic and Geometric Sequences
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7.1 Arithmetic and Geometric Sequences
Some properties of a geometric sequence
Let a be the first term, r be the common
ratio, Tn be the nth term and Sn be the sum
of the first n terms
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7.1 Arithmetic and Geometric Sequences
Some properties of a geometric sequence
(3) The sum of an infinite geometric sequence,
(4) a, b, c, are in arithmetic sequence, if and
only if b2 ac and b is called the geometric
mean.
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7.1 Arithmetic and Geometric Sequences
Some properties of a geometric sequence
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7.1 Arithmetic and Geometric Sequences
Let a1, a2, a3, ., an, be a sequence of real
numbers. The symbol
denotes the limit
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7.2 Harmonic Sequence (extension)
A sequence of non-zero numbers T1, T2, T3,is
called a harmonic sequence if and only if
are in arithmetic sequence.
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7.2 Harmonic Sequence (extension)
b is the harmonic mean of a and c if and only if
a, b, c are in harmonic sequence.
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P.242 Ex.7A
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7.3 The Method of Difference
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7.3 The Method of Difference
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7.3 The Method of Difference
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7.3 The Method of Difference
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7.3 The Method of Difference
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P.246 Ex.7B
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P.247 Ex.7C