SOLUTION: -39, -44, -49, -54, ... Find a35

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Question 1154055: -39, -44, -49, -54, ...
Find a35

Found 2 solutions by ikleyn, jim_thompson5910:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

This sequence represents an arithmetic progression.

Its first term is  a%5B1%5D = -39.  Its common difference is  d = -5.


The general formula for the n-th term of an arithmetic sequence is


    a%5Bn%5D = a%5B1%5D + %28n-1%29%2Ad.


In your case 


    a%5B35%5D = -39 + (-5)*(35-1),    or


    a%5B35%5D = -39 + (-5)*34 = -209.       ANSWER

Solved.

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For introductory lessons on arithmetic progressions see
    - Arithmetic progressions
    - The proofs of the formulas for arithmetic progressions
    - Problems on arithmetic progressions
    - Word problems on arithmetic progressions
in this site.

Also,  you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this online textbook under the topic "Arithmetic progressions".


Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

There are 2 ways to do this kind of problem.

The first method involves listing out 35 terms, each new term is found by subtracting 5 from the previous term (eg: -39 - 5 = -44, or -44-5 = -49, and so on). This is not recommended because there are so many terms. It's easy to lose track.

A much better method is to use the general nth term formula of an arithmetic sequence. That formula is
a%5Bn%5D+=+a%5B1%5D+%2B+d%28n-1%29
The first term is a%5B1%5D+=+-39. The common difference is d+=+-5 since we add on -5 each time, or subtract 5 each time, to get the next term.

So,
a%5Bn%5D+=+a%5B1%5D+%2B+d%28n-1%29

a%5Bn%5D+=+-39+%2B+%28-5%29%28n-1%29

a%5Bn%5D+=+-39-5n%2B5

a%5Bn%5D+=+-5n-34

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As a check, let's plug in a random positive integer between 1 and 4 to see if it matches up with the proper element in the sequence we're given. I'll do n = 3
a%5Bn%5D+=+-5n-34

a%5B3%5D+=+-5%283%29-34 Replace n with 3

a%5B3%5D+=+-15-34

a%5B3%5D+=+-49

The third term is -49

Sure enough, the third term of the given sequence {-39, -44, -49, -54, ...} is -49. I'll let you check the other values of n (1, 2 and 4) to see if things line up properly or not.

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Once we know the nth term formula, we plug in n = 35 to get the 35th term of the arithmetic sequence
a%5Bn%5D+=+-5n-34

a%5B35%5D+=+-5%2835%29-34 Replace n with 35

a%5B35%5D+=+-175-34

a%5B35%5D+=+-209

The 35th term is -209