Slide1 : Finite Sequences and Series
7.1 Arithmetic and Geometric Sequences : 7.1 Arithmetic and Geometric Sequences A sequence of numbers, T1, T2, T3,…is called an arithmetic sequence if and only if Tn+1 - Tn= d, for all n = 1, 2, …, where d is a constant (i.e. independent of n) called the common difference.
7.1 Arithmetic and Geometric Sequences : 7.1 Arithmetic and Geometric Sequences
7.1 Arithmetic and Geometric Sequences : 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
7.1 Arithmetic and Geometric Sequences : 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.
7.1 Arithmetic and Geometric Sequences : 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.
7.1 Arithmetic and Geometric Sequences : 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.
7.1 Arithmetic and Geometric Sequences : 7.1 Arithmetic and Geometric Sequences
7.1 Arithmetic and Geometric Sequences : 7.1 Arithmetic and Geometric Sequences
7.1 Arithmetic and Geometric Sequences : 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
7.1 Arithmetic and Geometric Sequences : 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.
7.1 Arithmetic and Geometric Sequences : 7.1 Arithmetic and Geometric Sequences Some properties of a geometric sequence :
7.1 Arithmetic and Geometric Sequences : 7.1 Arithmetic and Geometric Sequences Let a1, a2, a3, …., an,…
be a sequence of real numbers.
The symbol denotes the limit
7.2 Harmonic Sequence (extension) : 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.
7.2 Harmonic Sequence (extension) : 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.
P.242 Ex.7A : P.242 Ex.7A
7.3 The Method of Difference : 7.3 The Method of Difference
7.3 The Method of Difference : 7.3 The Method of Difference
7.3 The Method of Difference : 7.3 The Method of Difference
7.3 The Method of Difference : 7.3 The Method of Difference
7.3 The Method of Difference : 7.3 The Method of Difference