Mathematical tables: typical progression formulas
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$a_n:= a_1+\left( n-1\right)\cdot d$ ? The sum of first n elements of the arithmetic sequence Show source$S_n:=\frac{\left(2~ a_1+\left( n-1\right)\cdot d\right)}{2}\cdot n$ ? The sum of first n elements of the arithmetic sequence, if you know first and n-th elements Show source$S_n:=\frac{\left( a_1+ a_n\right)}{2}\cdot n$ ? 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$a_n:=\frac{\left( a_{n-1}+ a_{n+1}\right)}{2}$ ?

# Geometric progression (sequence)

 Name Formula Legend The N-th element of the geometric sequence Show source$a_n:= a_1+{ q}^{\left( n-1\right)}$ ? The sum of first n elements of the geometric sequence Show source$S_n:=\frac{ a_1\cdot\left(1-{ q}^{ n}\right)}{\left(1- q\right)}$ ? 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$a_n:=\sqrt{ a_{n-1}\cdot a_{n+1}}$ ?

# 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(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.