Solving Arithmetic Sequences


The following examples are problems involving arithmetic sequences. I included here several sample problems with their solutions.

Problem Number One :

If the first three terms of an arithmetic sequence are 2, 6 and 10, find the 40th term.

To solve the problem we use this formula for finding the nth term of an arithmetic sequence.

An = A + (n - 1) d

Where, An = is the nth term, in the case of our problem it is the 40th term

A = the first term of the sequence , in our problem it is 2.

n = number of terms, in our problem it is 40.

d = the interval of the terms, or the difference of the next term

from the previous term, To get d; d = 6 - 2 = 4.

Now, it is time to substitute the values to the formula for solving nth term where the 40th term is to be solved.

An = 2 + (40 - 1 ) 4

An = 2 + (39) 4

An = 2 + 156

An = 158.

The 40th term of the arithmetic sequence is 158.

Problem Number Two :

If the first term of an arithmetic sequence is -3 and the eighth term is 11, find d and write the first 10 terms of the sequence.

In this problem,

A = -3 n = 8 A8 = 11

If these values are substituted in the formula for An, we have

11 = -3 + (8 - 1) d

11 = -3 + 7d

14 = 7d

d = 2

The first ten terms are -3, -1, 1, 3, 5, 7, 9, 11, 13, 15


The sum of the first n terms of an arithmetic sequence with first term A and nth term An is;

Sn = n/2 (A + An) or this formula maybe rewritten as

Sn = n {(A+An)/2}

It can be remembered easily in this form as : "the number of terms multiplied by the mean value or average of the first and last terms."

For an arithmetic sequence with the first term A and common difference d, the sum of the first n terms is ;

Sn = n/2 { 2a + (n - 1 ) d }

Problem Number Three :

Find the sum of all the odd integers from 1 to 1111, inclusive.

Solution :

Since the odd integers 1, 3, 5, etc, taken in order from the arithmetic sequence with d = 2, we can first find n from the formula for the nth term;

  • 1111 = 1 + (n - 1) 2

1111 = 2n -1

1112 = 2n

n = 556

S = 556/2 ( 1 + 1111)

= 278 ( 1112)

= 309, 136

Problem Number Four :

If A = 4, n = 10, A10 = 49; find d and Sn.

Substituting the given values for A, n, and An in the formula:

An = A + (n - 1) d, we get

49 = 4 + (10 - 1 ) d

49 = 4 + 9d

45 = 9d

d = 5

By using Sn = n {( A + An)/2}, we have

S10 = 10 { (4 + 49 )/2} = 5 * 53 = 265

Source :

COLLEGE ALGEBRA (tenth edition ) by :

Paul K. Rees

Fred W. Sparks

Charles Sparks Rees

More by this Author

  • Solving Word Problems Involving Angular Velocity

      Solving Problems Involving Angular Velocity   Among the challenging problems I encounter  in Trigonometry are problems involving angular velocity. In this hub, I presented several problems...

  • Solving Word Problems Involving Exponential Function

    SolvingWord Problems Involving Exponential Function Exponential function is one of the most important concept in Algebra. In this hub I present several problems involving exponential function with their...

  • Solving Investment Problems

    Solving Investment Problems One of the most important applications of linear equations is found in solving investment problems. Investment problems use the Simple Interest formula I = Prt, where P =...

Comments 26 comments

siddhinfo profile image

siddhinfo 9 years ago from Goa (India)

Nice one really you have good write up.

jesusa 8 years ago

thank you.................................................................

morning star 8 years ago

Thank you very much , you are so beautiful and attractive. I love you .

gfj 8 years ago


Cory 8 years ago

this is the only site that explains it in a way i understand. thank you :)

Maria 7 years ago

Thank for the formula of the sums!

azil 7 years ago

hi ms. cristina, what if i was asked to complete the sequence, for example:

Complete the sequence 5 15 135 32805, if you don't mind, how should i answer that problem?

Hope you can help me understand about this!

Thanks in advance! =)

luke 6 years ago

hey azil its been 7 months...think u got blanked :)

salera gil john 6 years ago

thank you for the write up.... it helps me a lot as a future teacher major in mathematics...

sucks at mathalot:) 6 years ago

Thanks so much you are the only one who I found that could help me with this stuff. I have a test coming up and this was much appreciative

thanks a bundle :)

ate mhae 5 years ago

wow.... it is so very nice..! I love this one I understand now how to solve this simple math the arithmetic sequence... tnx.....

That white guy 5 years ago

Thanks this was incredibly helpfully for my 8th grade class homework now I can play black ops and reach lololololololololololol

You see I can say all these stupid things because I will never be on this excact web page again...@.@:@:)-729'd

nixie 5 years ago

thank u for the information!

chi 5 years ago

thank you!?

Rumy 5 years ago

Yes, Its help me a lot.


Hiruy 5 years ago

It's easy to learn sequence with out teacher ,so 10Q

Nevaeh 4 years ago

Thaanks buddy : )

Kendall 4 years ago


relente 4 years ago

thank uo

jonna mae 4 years ago

its easyto learn arithmetic sequence

edjeboy desacula 4 years ago

thanks God........... God bless

john 4 years ago


mjean 4 years ago

. tnx . u help me have my assignments and proje


DIWATANIANS 4 years ago


i think this is cool

rish 3 years ago

Thank u vry max...

    Sign in or sign up and post using a HubPages Network account.

    0 of 8192 characters used
    Post Comment

    No HTML is allowed in comments, but URLs will be hyperlinked. Comments are not for promoting your articles or other sites.

    Click to Rate This Article