Tables show common formulas helpful when you performing sequences related tasks such as sum of first n elements of arithmetic sequence or calculation arbitral element of geometric sequence.

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Arithmetic progression (sequence)#

Name	Formula	Legend
The N-th element of the arithmetic sequence	Show source $a_n=a_1+\left(n-1\right) \cdot d$	$a_n$ - the n-th element of the sequence, $a_1$ - the first element of the sequence, d - common difference of arithmetic sequence (the difference of successive sequence elements: $a_{n+1} - a_n$ ).
The sum of first n elements of the arithmetic sequence	Show source $S_n=\frac{2~a_1+\left(n-1\right) \cdot d}{2} \cdot n$	$S_n$ - the sum of first n elements of the sequence, $a_1$ - the first element of the sequence, d - common difference of arithmetic sequence (the difference of successive sequence elements: $a_{n+1} - a_n$ ).
The sum of first n elements of the arithmetic sequence, if you know first and n-th elements	Show source $S_n=\frac{a_1+a_n}{2} \cdot n$	$S_n$ - the sum of first n elements of the sequence, $a_1$ - the first element of the sequence, $a_n$ - the n-th element of the sequence.
The common difference of arithmetic sequence	Show source $d=a_{n+1}-a_n$	d - common difference of arithmetic sequence (the difference of successive sequence elements: $a_{n+1} - a_n$ ), $a_{n+1}$ - the (n+1)-th element of the sequence (the element just after $a_n$ ), $a_n$ - the n-th element of the sequence.
The relationship between three consecutive elements of a arithmetic sequence	Show source $a_n=\frac{a_{n-1}+a_{n+1}}{2}$	$a_n$ - the n-th element of the sequence, $a_{n+1}$ - the (n+1)-th element of the sequence (the element just after $a_n$ ), $a_{n-1}$ - (n-1)-th element of the sequence (the element just before $a_n$ ).

Geometric progression (sequence)#

Name	Formula	Legend
The N-th element of the geometric sequence	Show source $a_n=a_1+q^{n-1}$	$a_n$ - the n-th element of the sequence, $a_1$ - the first element of the sequence, q - common ratio of geometric sequence (ratio between succesive sequence elements: $a_{n+1} / a_n$ ).
The sum of first n elements of the geometric sequence	Show source $S_n=\frac{a_1 \cdot \left(1-q^{n}\right)}{1-q}$	$S_n$ - the sum of first n elements of the sequence, $a_1$ - the first element of the sequence, q - common ratio of geometric sequence (ratio between succesive sequence elements: $a_{n+1} / a_n$ ).
The common ratio of geometric sequence	Show source $q=\frac{a_{n+1}}{a_n}$	q - common ratio of geometric sequence (ratio between succesive sequence elements: $a_{n+1} / a_n$ ), $a_{n+1}$ - the (n+1)-th element of the sequence (the element just after $a_n$ ), $a_n$ - the n-th element of the sequence.
The relationship between three consecutive elements of a geometric sequence	Show source $a_n=\sqrt{a_{n-1} \cdot a_{n+1}}$	$a_n$ - the n-th element of the sequence, $a_{n+1}$ - the (n+1)-th element of the sequence (the element just after $a_n$ ), $a_{n-1}$ - (n-1)-th element of the sequence (the element just before $a_n$ ).

Some facts#

Numerical sequence (sometimes also called numerical progression) is a function whose arguments are natural numbers (1, 2, 3, etc.):

$\begin{alignedat}{4} f(1) & = a_1 & = & \text{ the first term of the sequence},\\ f(2) & = a_2 & = & \text{ the second term of the sequence},\\ f(3) & = a_3 & = & \text{ the third term of the sequence},\\ ...\\ f(n-1) & = a_{n-1} & = & \text{ the (n-1)-th term of the sequence},\\ f(n) & = a_{n} & = & \text{ the n-th term of the sequence},\\ f(n+1) & = a_{n+1} & = & \text{ the (n+1)-th term of the sequence},\\ \text{etc.} \end{alignedat}$
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.

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.

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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