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This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !
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Arithmetic progression (sequence)
Name  Formula  Legend 
The Nth element of the arithmetic sequence  Show source$a_n= a_1+\left( n1\right)\cdot d$ 

The sum of first n elements of the arithmetic sequence  Show source$S_n=\frac{2~ a_1+\left( n1\right)\cdot d}{2}\cdot n$ 

The sum of first n elements of the arithmetic sequence, if you know first and nth elements  Show source$S_n=\frac{ a_1+ a_n}{2}\cdot n$ 

The common difference of arithmetic sequence  Show source$d= a_{n+1} a_n$ 

The relationship between three consecutive elements of a arithmetic sequence  Show source$a_n=\frac{ a_{n1}+ a_{n+1}}{2}$ 

Geometric progression (sequence)
Name  Formula  Legend 
The Nth element of the geometric sequence  Show source$a_n= a_1+{ q}^{ n1}$ 

The sum of first n elements of the geometric sequence  Show source$S_n=\frac{ a_1\cdot\left(1{ q}^{ n}\right)}{1 q}$ 

The common ratio of geometric sequence  Show source$q=\frac{ a_{n+1}}{ a_n}$ 

The relationship between three consecutive elements of a geometric sequence  Show source$a_n=\sqrt{ a_{n1}\cdot a_{n+1}}$ 

Some facts
 Numerical sequence (sometimes also called numerical progression) is a function whose arguments are natural numbers (1, 2, 3, etc.):
$\begin{alignedat}{4} f(1) & = a_1 & = & \text{ the first term of the sequence},\\ f(2) & = a_2 & = & \text{ the second term of the sequence},\\ f(3) & = a_3 & = & \text{ the third term of the sequence},\\ ...\\ f(n1) & = a_{n1} & = & \text{ the (n1)th term of the sequence},\\ f(n) & = a_{n} & = & \text{ the nth term of the sequence},\\ f(n+1) & = a_{n+1} & = & \text{ the (n+1)th term of the sequence},\\ \text{etc.} \end{alignedat}$  The sequence differs from the set in that its elements are ordered (the order of the elements matter).
 The arithmetic sequence is the sequence in which each successive element differs from the previous one by a fixed value d:
$a_{n+1} = a_n + d$where:
 $a_n$  arbitrarily selected term,
 $a_{n+1}$  the term just after $a_n$,
 $d$  common difference of arithmetic sequence.
 $a_n$  arbitrarily selected term,
 If you want to learn more about the arithmetic sequence, check our other calculator: Arithmetic sequence.
 The geometric sequence is a sequence in which each successive element is r times greater than the previous one:
$a_{n+1} = a_n \cdot r$where:
 $a_n$  arbitrarily selected term,
 $a_{n+1}$  the term just after $a_n$,
 $r$  common ratio of geometric sequence.
 $a_n$  arbitrarily selected term,
 If you want to learn more about the geometric sequence, check our other calculator: Geometric sequence.
 In addition to the numerical sequence, we can consider sequences composed of other mathematical objects, e.g. functions. In this case, we would talk about function sequences.
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