Arithmetic progression calculator
Calculator for tasks related to arithmetic sequences such as sum of n first elements or calculation of selected n-th term of the progression.

Beta version#

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Symbolic algebra

ⓘ Hint: This calculator supports symbolic math. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. If you still don't sure how to make your life easier using symbolic algebra check out our another page: Symbolic calculations

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Calculations data - enter values, that you know here#

The n-th element of the sequence (ana_n)
The sum of first n elements of the sequence (SnS_n)
Common difference of arithmetic sequence (d)
(the difference of successive sequence elements: an+1ana_{n+1} - a_n)
The first element of the sequence (a1a_1)
The (n+1)-th element of the sequence (an+1a_{n+1})
(n-1)-th element of the sequence (an1a_{n-1})

Result: the n-th element of the sequence (ana_n)#

Used formulaShow sourcean=a1+(n1)da_n=a_1+\left(n-1\right) \cdot d
ResultShow source11
Numerical resultShow source11
Result step by step
1Show source1+(11)11+\left(1-1\right) \cdot 1Multiply by oneAny number multiplied by one (1) gives the same number: a1=1a=aa \cdot 1 = 1 \cdot a = a
2Show source1+111+1-1Simplify arithmetic-
3Show source212-1Simplify arithmetic-
4Show source11ResultYour expression reduced to the simplest form known to us.
Numerical result step by step
1Show source11The original expression-
2Show source11ResultYour expression reduced to the simplest form known to us.

Some facts#

  • 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
    • ana_n - arbitrarily selected term,
    • an+1a_{n+1} - the term just after ana_n,
    • dd - common difference of arithmetic sequence.
  • The above formula should be understood as follows: if I know some element of the arithmetic sequence (ana_n) and its difference (rr), then I can calculate the next one (an+1a_{n + 1}).
  • The above formula can be also formulated as below:
    an=an1+da_{n} = a_{n-1} + d
    • ana_n - arbitrarily selected term (except the first one: n1n \neq 1),
    • an1a_{n-1} - the term just before ana_n,
    • dd - the common difference of arithmetic sequence.
  • Above alternative formula should be understood as follows: if I want to calculate some selected element of the arithmetic sequence (ana_{n}), then I need to know the previous one (an1)a_{n-1})) and the common difference (dd).
  • It is worth noting that the second formula does not work for the first element (a1a_1). This is because the first term as the only one does not have the previous element.
  • In order to uniquely define the arithmetic sequence, it is enough to know two values:
    • the first term a1a_1,
    • and the difference between two consecutive terms dd, so called common difference of arithmetic sequence:
      d=an+1and = a_{n+1} - a_n
  • Arithmetic sequence is sometimes called an arithmetic progression.
  • If you are interested in the properties of sequences, then you can check out our other calculators:

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