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

What do you want to calculate today?#

Choose a scenario that best fits your needs

I know the first element of the sequence (

a_1

), common ratio of geometric sequence (q) and n and want to calculate the n-th element of the sequence (

a_n

)I know the first element of the sequence (

a_1

), common ratio of geometric sequence (q) and n and want to calculate the sum of first n elements of the sequence (

S_n

)I know the (n+1)-th element of the sequence (

a_{n+1}

) and the n-th element of the sequence (

a_n

) and want to calculate common ratio of geometric sequence (q)I know the (n+1)-th element of the sequence (

a_{n+1}

) and (n-1)-th element of the sequence (

a_{n-1}

) and want to calculate the n-th element of the sequence (

a_n

)

Calculations data - enter values, that you know here#

The n-th element of the sequence ( $a_n$ )		=>
The sum of first n elements of the sequence ( $S_n$ )		=>
Common ratio of geometric sequence (q) (ratio between succesive sequence elements: $a_{n+1} / a_n$ )		<=
The first element of the sequence ( $a_1$ )		<=
N		<=
The (n+1)-th element of the sequence ( $a_{n+1}$ )		=>
(n-1)-th element of the sequence ( $a_{n-1}$ )		=>

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

Summary

Used formula

Show source

a_n=a_1+q^{n-1}

Result

Show source

2

Numerical result

Show source

2

Result step by step

1	Show source $1+1^{1-1}$	Power of one number	The number one (1) raised to any power gives one (1). $1^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{\text{n razy}} = 1$
2	Show source $1+1$	Simplify arithmetic	-
3	Show source $2$	Result	Your expression reduced to the simplest form known to us.

Numerical result step by step

1	Show source $2$	The original expression	-
2	Show source $2$	Result	Your 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:

$a_{n+1} = a_n \cdot r$
where:
- $a_n$ - arbitrarily selected term,
- $a_{n+1}$ - the term just after $a_n$ ,
- $r$ - common ratio of geometric sequence.
The above formula should be understood as follows: if I know some element of the geometric sequence ( $a_n$ ) and its common ratio ( $d$ ), then I can calculate the next one ( $a_{n + 1}$ ).
we can also define a geometric sequence in a slightly different way:

$a_{n} = a_{n-1} \cdot r$
where:
- $a_n$ - arbitrarily selected term (except the first one: $n \neq 1$ ),
- $a_{n-1}$ - the term just before $a_n$ ,
- $r$ - 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 ( $a_{n}$ ), then I need to know the previous one ( $a_{n-1})$ ) and the common ration ( $r$ ).
It is worth noting that the second formula does not work for the first element ( $a_1$ ). 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 $a_1$ ,
- and the ratio of two consecutive terms $r$ , so called common ratio of geometric sequence:
  
  $r = \dfrac{a_{n+1}}{a_n}$
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:
- Arithmetic sequence,
- Numerical sequences.