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$a_n= a_1+\left( n-1\right)\cdot d$ $a_n$ - 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.