An item in a sequence is called a term. The nth term of a sequence is denoted with - the first term is . A sequence can be arithmetic (common difference) or geometric (common ratio).
A series is the sum of the first n terms of a sequence. A series can be arithmetic (common difference) or geometric (common ratio).
recurrence / explicit
recurrence
A recurrence relation like is a relation that defines a sequence recursively.
It means each term is defined using previous terms.
A recurrence relation must have initial conditions - these are specific values of the sequence that are defined directly.
Consider the sequence 3,5,7,9… - The recurrence relation is , .
Consider the sequence 1,1,2,3,5,8,13… - The recurrence relation is , , .
explicit
An explicit formula like defines every nth term directly as a function of n.
a is the first term
d is the common difference (calculated with the common difference between three terms)
Consider the sequence 3,5,7,9… - The explicit formula is .
Consider the sequence 2,5,10,17,26… - The explicit formula is .
arithmetic sequence
id: 1744589592928
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Arithmetic sequence is ==$t_n=a+(n-1)d$==a is the first term
d is the common difference (calculated with the common difference between three terms)
arithmetic series
For an arithmetic sequence, the arithmetic series is a sum of some terms.
There are two equations we use:
id: 1744589592964
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Arithmetic series is ==$S_n=\dfrac{n}{2}[2a+(n-1)d]$==
l is the last term
geometric sequence
id: 1744589592988
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Geometric sequence is ==$t_n=ar^{n-1}$==a is first term
r is common ratio (calculated with the common ratio between three terms)
geometric series
id: 1744589593014
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Geometric series is ==$S_n=\dfrac{a(r^n-1)}{r-1}$==infinite geometric series
id: 1744589593067
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Infinite geometric series is ==$S_\infty=\dfrac{a}{1-r}, |r|<1$==