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)da_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)d2nS_n=\frac{2~a_1+\left(n-1\right) \cdot d}{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+an2nS_n=\frac{a_1+a_n}{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+1and=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+12a_n=\frac{a_{n-1}+a_{n+1}}{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+qn1a_n=a_1+q^{n-1}
  • 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)1qS_n=\frac{a_1 \cdot \left(1-q^{n}\right)}{1-q}
  • 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+1anq=\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+1a_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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