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Arithmetic Sequence Calculator
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Arithmetic Sequence Calculator is an online tool used to find nth term and sum of the sequence.

Arithmetic sequence is a sequence of numbers such that the distinction between two consecutive members of the sequence as the same common difference. 
 

Step by Step Calculation

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Step 1 :  

The formula for the nth term an of an infinite arithmetic sequence with a common difference d and a first term a1 is given by




                 an = a1 + (n - 1 )d




The sum sn of the first n terms of an infinite arithmetic sequence is defined by




                 sn = a1 + a2 + a3 + ... + an



and is a1 is given by



                 sn = n (a1 + an) / 2


 


Where


 


  a1 = first number of the sequence




          d = common difference between the sequence




       an = nth term of the sequence




          n = total number of value in the sequence.


 


       Sn = Sum of nth term of the sequence



Step 2 :  

Put the values in the formulas and calculate it further.



Example Problems

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  1. Find the sum of all positive integers, from 27 to 4590 inclusive, which are divisible by 27.


    Step 1 :  

    Given: Sequence of first few numbers of positive integers divisible by 27 are given by 27, 54, 81...




    First term of the sequence 'a' = 27


     


    Common difference 'd' = 27.


     


    We need to know the rank of the term 4590.




    We use the following formula for the nth term




    an = a1 + (n - 1 )d


     


    The sum sn of the first n terms of an infinite arithmetic sequence is defined by




                     sn = a1 + a2 + a3 + ... + an




    and is a1 is given by




                     sn = n (a1 + an) / 2



    Step 2 :  

    Put the values in the formula and solve it further.


     


    an = a1 + (n - 1 )d




    4590 = a1 + (n - 1 )d




    Substitute a1 and d by their values




            4590 = 27 + 27(n - 1)




    Solve for n to obtain




            n = 170




     4590 is the 170th term, we can use the following formula to find sum




            sn = n (a1 + an) / 2




            s170 = 170 (27 + 4590) / 2 = 392445.



    Answer  :  

    Term at position 170 = 4590

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    Sum of all terms till position = 392445.



  2. Find the sum of all positive integers, from 23 to 3680 inclusive, which are divisible by 23.


    Step 1 :  

    Given: Sequence of first few numbers of positive integers divisible by 23 are given by 23, 46, 69...


     



    First term of the sequence 'a' = 23


     


    Common difference 'd' = 23


     


    We need to know the rank of the term 3680.




    We use the following formula for the nth term




    an = a1 + (n - 1 )d




    The sum sn of the first n terms of an infinite arithmetic sequence is defined by




                     sn = a1 + a2 + a3 + ... + an




    and is a1 is given by




                    
    sn = n (a1 + an) / 2



    Step 2 :  

    Put the values in the formula and solve it further.


     


    an = a1 + (n - 1 )d


     


    3680 = a1 + (n - 1 )d




    Substitute a1 and d by their values




            3680 = 23 + 23(n - 1)




    Solve for n to obtain




            n = 160




    3680 is the 160th term, we can use the following formula to find sum




            sn = n (a1 + an) / 2




            s160 = 160 (23 + 3680) / 2 = 296240.



    Answer  :  

    Term at position 170 = 3680

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    Sum of all terms till position = 296240.



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