# Solving Arithmetic Sequences

**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

**SUM OF AN ARITHMETIC SEQUENCE **

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

- 4
Solving Geometry Problems Involving System of Equations In Two Variables Algebra has also found wide applications in Geometry. Among its common problems are problems involving perimeters and angles. In this hub I...

- 4
Solving Word Problems Involving Chebyshev’s Theorem Chebyshev’s Theorem states that the proportion or percentage of any data set that lies within k standard deviation of the mean where k is any...

- 3
Solving Work Problems Involving System of Linear Equations In this hub I presented several challenging work problems involving system of linear equations complete with solution. I hope this hub will be useful...

## Comments 26 comments

Nice one really you have good write up.

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

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

nice

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

Thank for the formula of the sums!

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! =)

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

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

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 :)

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

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

thank u for the information!

thank you!?

Yes, Its help me a lot.

Thanks

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

Thaanks buddy : )

Thanks!

thank uo

its easyto learn arithmetic sequence

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

hi```

. tnx . u help me have my assignments and proje

cts

WHOAH.... THANKS FOR HELPING MY ASSIGNMENT!

i think this is cool

Thank u vry max...

26