Geometric progression calculator
Calculator for tasks related to geometric 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 ratio of geometric sequence (q)
(ratio between succesive sequence elements: an+1/ana_{n+1} / a_n)
<=
The first element of the sequence (a1a_1)
<=
N
<=
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)#

Summary
Used formulaShow sourcean=a1+qn1a_n=a_1+q^{n-1}
ResultShow source22
Numerical resultShow source22
Result step by step
1Show source1+1111+1^{1-1}Power of one numberThe number one (1) raised to any power gives one (1). 1n=1111n razy=11^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{\text{n razy}} = 1
2Show source1+11+1Simplify arithmetic-
3Show source22ResultYour expression reduced to the simplest form known to us.
Numerical result step by step
1Show source22The original expression-
2Show source22ResultYour expression reduced to the simplest form known to us.

Some facts#

  • 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.
  • The above formula should be understood as follows: if I know some element of the geometric sequence (ana_n) and its common ratio (dd), then I can calculate the next one (an+1a_{n + 1}).
  • we can also define a geometric sequence in a slightly different way:
    an=an1ra_{n} = a_{n-1} \cdot r
    where:
    • ana_n - arbitrarily selected term (except the first one: n1n \neq 1),
    • an1a_{n-1} - the term just before ana_n,
    • rr - the common ratio of geometric sequence.
  • Above alternative formula should be understood as follows: if I want to calculate some selected element of the geometric sequence (ana_{n}), then I need to know the previous one (an1)a_{n-1})) and the common ration (rr).
  • 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 geometric sequence, it is enough to know two values:
    • the first term a1a_1,
    • and the ratio of two consecutive terms rr, so called common ratio of geometric sequence:
      r=an+1anr = \dfrac{a_{n+1}}{a_n}
  • Geometric sequence is sometimes called a geometric progression.
  • If you are interested in the properties of sequences, then you can check out our other calculators:

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