Arithmetic and Geometric Sequences
8:41 PM
Muhammad Yusuf
, Posted in
Arithmetic Sequences
An
arithmetic sequence is a sequence where the difference between any consecutive
pairs of numbers in the sequence is a constant. This difference is called the common
difference.
For
example: 1, 3, 5, 7, 9, ... is an arithmetic sequence with first term 1 and common difference 2.
Are
any of the sequences above arithmetic sequences? Which ones?
The
terms in an arithmetic sequence with first term a and
common difference d are:
a1 = a, a2 = a + d, a3 = (a + d) + d = a + 2d, a4 = (a + 2d) + d = a + 3d,
...
In
general, the nth term of an arithmetic sequence with first term a and
common difference d is:
an = a + (n - 1)d
Eg.,
in the sequence 1, 3, 5, 7, 9, ..., the 5th term is a5 = 1 + (5 - 1)2 = 9
2.
What is the 23rd term in the sequence 18, 11, 4, -3, ...?
3. If
the 4th term of an arithmetic sequence is 11 and the 8th is 31, determine the
100th.
Geometric Sequences
A
geometric sequence is a sequence where there is the same ratio between any consecutive
terms. This ratio is called the common ratio.
For
example: 1, 2, 4, 8, 16, ... is an infinite geometric sequence with first term 1 and common ratio 2.
Are
any of the sequences above geometric sequences? Which ones?
The
terms in a geometric sequence with first term a and
common ratio r are:
a1 = a, a2 = ar, a3 = (ar)r = ar2, a4 = (ar2)r = ar3, ...
In
general, the nth term of a geometric sequence with first term a and
common ratio r is:
an = arn-1
Eg.,
in the sequence 1, 2, 4, 8, 16, ..., the 5th term is a5 = 1 × (2)5-1 = 16
4.
State the first 4 terms of the geometric sequence with general term an = 2 × (-3)n-2
5.
What is the 11th term of the sequence 9, 3, 1, ⅓ , ...?
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