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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)

NameFormulaLegend
The N-th element of the arithmetic sequenceShow sourcean:=a1+(n1)d a_n:= a_1+\left( n-1\right)\cdot d
  • ana_n - the n-th element of the sequence,
  • a1a_1 - the first element of the sequence,
  • d - common difference of arithmetic sequence (the difference of successive sequence elements: an+1ana_{n+1} - a_n).
The sum of first n elements of the arithmetic sequenceShow sourceSn:=(2 a1+(n1)d)2n S_n:=\frac{\left(2~ a_1+\left( n-1\right)\cdot d\right)}{2}\cdot n
  • SnS_n - the sum of first n elements of the sequence,
  • a1a_1 - the first element of the sequence,
  • d - common difference of arithmetic sequence (the difference of successive sequence elements: an+1ana_{n+1} - a_n).
The sum of first n elements of the arithmetic sequence, if you know first and n-th elementsShow sourceSn:=(a1+an)2n S_n:=\frac{\left( a_1+ a_n\right)}{2}\cdot n
  • SnS_n - the sum of first n elements of the sequence,
  • a1a_1 - the first element of the sequence,
  • ana_n - the n-th element of the sequence.
The common difference of arithmetic sequenceShow sourced:=an+1an d:= a_{n+1}- a_n
  • d - common difference of arithmetic sequence (the difference of successive sequence elements: an+1ana_{n+1} - a_n),
  • an+1a_{n+1} - the (n+1)-th element of the sequence (the element just after ana_n),
  • ana_n - the n-th element of the sequence.
The relationship between three consecutive elements of a arithmetic sequenceShow sourcean:=(an1+an+1)2 a_n:=\frac{\left( a_{n-1}+ a_{n+1}\right)}{2}
  • ana_n - the n-th element of the sequence,
  • an+1a_{n+1} - the (n+1)-th element of the sequence (the element just after ana_n),
  • an1a_{n-1} - (n-1)-th element of the sequence (the element just before ana_n).

Geometric progression (sequence)

NameFormulaLegend
The N-th element of the geometric sequenceShow sourcean:=a1+q(n1) a_n:= a_1+{ q}^{\left( n-1\right)}
  • ana_n - the n-th element of the sequence,
  • a1a_1 - the first element of the sequence,
  • q - common ratio of geometric sequence (ratio between succesive sequence elements: an+1/ana_{n+1} / a_n).
The sum of first n elements of the geometric sequenceShow sourceSn:=a1(1qn)(1q) S_n:=\frac{ a_1\cdot\left(1-{ q}^{ n}\right)}{\left(1- q\right)}
  • SnS_n - the sum of first n elements of the sequence,
  • a1a_1 - the first element of the sequence,
  • q - common ratio of geometric sequence (ratio between succesive sequence elements: an+1/ana_{n+1} / a_n).
The common ratio of geometric sequenceShow sourceq:=an+1an q:=\frac{ a_{n+1}}{ a_n}
  • q - common ratio of geometric sequence (ratio between succesive sequence elements: an+1/ana_{n+1} / a_n),
  • an+1a_{n+1} - the (n+1)-th element of the sequence (the element just after ana_n),
  • ana_n - the n-th element of the sequence.
The relationship between three consecutive elements of a geometric sequenceShow sourcean:=an1an+1 a_n:=\sqrt{ a_{n-1}\cdot a_{n+1}}
  • ana_n - the n-th element of the sequence,
  • an+1a_{n+1} - the (n+1)-th element of the sequence (the element just after ana_n),
  • an1a_{n-1} - (n-1)-th element of the sequence (the element just before ana_n).

Some facts

  • Numerical sequence (sometimes also called numerical progression) is a function whose arguments are natural numbers (1, 2, 3, etc.):
    f(1)=a1= the first term of the sequence,f(2)=a2= the second term of the sequence,f(3)=a3= the third term of the sequence,...f(n1)=an1= the (n-1)-th term of the sequence,f(n)=an= the n-th term of the sequence,f(n+1)=an+1= the (n+1)-th term of the sequence,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:
    an+1=an+da_{n+1} = a_n + d
    where:
    • ana_n - arbitrarily selected term,
    • an+1a_{n+1} - the term just after ana_n,
    • dd - 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:
    an+1=anra_{n+1} = a_n \cdot r
    where:
    • ana_n - arbitrarily selected term,
    • an+1a_{n+1} - the term just after ana_n,
    • rr - 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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