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#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

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 difference of arithmetic sequence (d) 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 difference of arithmetic sequence (d) and n and want to calculate the sum of first n elements of the sequence (

S_n

)I know the first element of the sequence (

a_1

), the n-th element of the sequence (

a_n

) 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 difference of arithmetic sequence (d)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 difference of arithmetic sequence (d) (the difference of successive 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+\left(n-1\right) \cdot d

Result

Show source

1

Numerical result

Show source

1

Result step by step

1	Show source $1+\left(1-1\right) \cdot 1$	Multiply by one	Any number multiplied by one (1) gives the same number: $a \cdot 1 = 1 \cdot a = a$
2	Show source $1+1-1$	Simplify arithmetic	-
3	Show source $2-1$	Simplify arithmetic	-
4	Show source $1$	Result	Your expression reduced to the simplest form known to us.

Numerical result step by step

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

$a_{n+1} = a_n + d$
where:
- $a_n$ - arbitrarily selected term,
- $a_{n+1}$ - the term just after $a_n$ ,
- $d$ - common difference of arithmetic sequence.
The above formula should be understood as follows: if I know some element of the arithmetic sequence ( $a_n$ ) and its difference ( $r$ ), then I can calculate the next one ( $a_{n + 1}$ ).
The above formula can be also formulated as below:

$a_{n} = a_{n-1} + d$
where:
- $a_n$ - arbitrarily selected term (except the first one: $n \neq 1$ ),
- $a_{n-1}$ - the term just before $a_n$ ,
- $d$ - 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 ( $a_{n}$ ), then I need to know the previous one ( $a_{n-1})$ ) and the common difference ( $d$ ).
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 arithmetic sequence, it is enough to know two values:
- the first term $a_1$ ,
- and the difference between two consecutive terms $d$ , so called common difference of arithmetic sequence:
  
  $d = a_{n+1} - a_n$
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:
- Geometric sequence,
- Numerical sequences.