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Contents
Sequences & Series
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Sequences & Series
A sequence is a number pattern in a
definite order following a certain rule.
Examples of sequences:
1) 1, 2, 3, 4, 5, 6, 7,
...
add 1 to the preceding
term
2) 2, 4, 7, 11, 16, 23,
31. add
2 to the preceding term, add 3 to the next term, etc
3) 1, 1, 2, 3, 5, 8, 13, 21,
34,... add
the two preceding terms together
The last sequence is known as the
Fibionacci sequence, as discovered by Leonardo of Pisa. This sequence occurs in
nature, and Leonardo of Pisa derived it by studying the mating patterns of
rabbits.
Sequences are usually denoted by:
T1, T2, T3, T4, ...
A series is a
sum of terms in a sequence. It can be denoted by
T1+ T2+ T3+ T4+ ...
Using the above sequences, we have
the following series:
1) 1 + 2 + 3 + 4 + 5 + 6 + 7 +...
2) 2 + 4 + 7 + 11 + 16 + 23 + 31.
3) 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21
+ 34 +...
Sequences and series can be finite
or infinite. A finite sequence/series is one that eventually comes to an end,
like the second one in the examples above. Infinite sequences/series are those
that continue indefinitely, such as the first in the example as well as the
Fibionacci sequence.
Arithmetic & Geometric
Progressions
Arithmetic and geometric
progressions, commonly abbreviated to A.P. and G.P. respectively, are two forms
of sequences. Their definitions are given later in this section. The
applications of these sequences are more theoretical than practical, though the
idea can be used to calculate values (distances, length, cost, etc) for
practical situations whereby sequences in the form of A.P./G.P. are employed.
One example is the second example in our section of Sum to Infinity.
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