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Arithmetic Progression used to find the nth term

Updated on October 4, 2010

Arithmetic Progression as requested by baig772

 Arithmetic Progression

An arithmetic progression or arithmetic sequence are a sequence of numbers that increase or decrease by a common difference.

Solving arithmetic progression is interesting since we can find the nth term of a particular sequence in a much easier way.

Within this hub we shall cover the steps involved in progression solving for;

  • Finding the nth term
  • Find a chosen term as defined by the question
  • Finding the A.P.(arithmetic progression ) itself

The general form of an arithmetic sequence is;

a,  a + d,  a + 2d,  a + 3d,  a + 4d...

Where the  a   =  first term (number)

                   d  =  common difference

To find the nth term of an arithmetic sequence we use the following formula;

tn  =  a  +  (n - 1)  d

When we add or subtract any constant number with all the terms of the sequence, the arithmetic sequence remains an arithmetic sequence.


6, 8, 10, 12, 14, 16, 18….. is an A.P with a common difference 2.

Add 4 with all the terms, and we get,

10, 12, 14, 16, 18, 20, 22…. this is also an A.P with a common difference 2.


When we multiply or divide by a non-zero constant with the terms of a sequence, the arithmetic sequence remains an arithmetic sequence.


10, 20, 30, 40, 50, 60, 70…. Is an A.P with a common difference 10.

Multiply all the terms by 2 , and we get,

20, 40, 60, 80, 100, 120, 140…. this is also an A.P with a common difference 20.


Finding the nth term


( i )  The sequence terms of an A.P. are  1, 12, 23, 34 find the nth term


t=  a  +  (n - 1) d

a   =  1                         ( where a = the first term)

d   =  12 - 1 = 11        (where d = common difference)

t=  1 + (n - 1)  d

t=  1 + (n - 1) 11

t=  1 + 11n -  11

t=  11n - 10  (nth term)


( ii )  Find the 11th term of the sequence 1, 12, 23, 34


a = 1                     (first term)

d = 12 - 1 = 11    (common difference)

n = 11                  (number of term required)


tn  =  a  +  ( n - 1)  d

t11 = 1  +  (11 - 1)  x  11

      = 1  +   (10   x   11)

      = 1  +  110

t11 =  111


(iii)  Find the A.P. of a sequence when the 7th term is 67

                                                              and the 16th term is 166


Consider the A.P. in the form a,  a + d,  a + 2d,  a + 3d,  a + 4d......


 t=  a  +  6d  =  67

t16 =  a  + 15d = 166

t7 - t16  =   a  +  6d  =  67

             = - a  - 15d = -166

             =        -  9d =  -99        

 -9d  =  -99   =   11

-9          -9


d  =  11


Put   "d = 11"  into the t7 equation


a  +      6d        =  67

a  =  (6 x 11)   =  67

a  =      66        =  67

a  =     67  -  66  =  1

a  =  1

t1 =   a  =  1

t2 =   a  +  d  =  1  +    11        =  12

t3 =   a  + 2d = 1  + (2 x 11)   =  23

t4 =   a  + 3d = 1  = (3 x 11)   =  34


So terms for this A.P. sequence are;   1,   12,   23,   34


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    • t.elia profile imageAUTHOR


      5 years ago from Northern Ireland

    • profile image


      6 years ago

      Thank You!!!, sa mga aNsWer... heheheh

    • profile image

      jonas moyo 

      6 years ago

      hi guys you made my day, i did this 8 yrs ago and now you have refreshad my memory

    • t.elia profile imageAUTHOR


      8 years ago from Northern Ireland

      Thanks for the acknowledgement claree. Glad to be of help to you.

    • profile image


      8 years ago

      Thanks- this really helped me in my exam studies

    • t.elia profile imageAUTHOR


      8 years ago from Northern Ireland

      I couldn't agree more billy, I really enjoy maths although Im a chef by trade. Thankyou for your kind remarks really appreciated.Will pop over to your site later as I find some of your topics a great read and very interesting.


    • billyaustindillon profile image


      8 years ago

      You have some great meths hubs. I do a lot of research with fractals, fibonacci and other works like Gann, Elliott and Murrey in the financial markets. Maths and nature are what makes those studies work and viable. There is nothing as clinical as maths.

    • t.elia profile imageAUTHOR


      8 years ago from Northern Ireland


      Hope this helps and thanks for the request



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