Question 1121709
.
Actually, there are &nbsp;<U>TENS</U> &nbsp;arithmetic progressions satisfying the given conditions.



So the question and the problem, as they formulated, posted and presented, &nbsp;<U><I>MAKE &nbsp;NO &nbsp;SENSE</I></U>.



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<U>Comment from student</U> : Can you also help me out with this last question? 
Write down the first three terms of a arithmetic series who’s sum is 1212 The series must have a minimum of 13 terms. 
The first term and the common difference are integers.
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<U>My response</U> :


<pre>
The formula for the sum of the first n terms of an arithmetic progression is


    {{{S[n]}}} = {{{((a[1]+a[n])/2)*n}}}.


So, in your case it should be


    {{{((a[1]+a[n])/2)*n}}} = 1212,   or   {{{(a[1]+a[n])*n}}} = 2424.


Notice that 2424 = 24*101, and 101 is a prime number.


It means that "n" must divide 101 or must divide some multiple of 101.



Now the first example of such arithmetic progression is THIS:

    101 terms, each equal to 12.

    It is arithmetic progression consisting of 101 equal terms;  each term is equal to 12; the common difference is 0 (zero).



The second example is 

    202 terms, each equal to 6.

    It is arithmetic progression consisting of 202 equal terms;  each term is equal to 6; the common difference is 0 (zero).



Having these two examples, you can easily construct other similar examples with the common difference of 0.



I will help you to construct one more example with the non-zero common difference.

    51-th term is 12;

    50-th term is 11;  52-th term is 13;

    49-th term is 10;  53-th term is 14.

    . . .   and  so  on  . . . . . . . . 


    1-st term is 12-50 = - 38;  101-th term is 12+50 = 62.


Notice that each pair of the terms in each line has half the sum of 12 (!)


Thus, the progression starts from -38; has the common difference of 1; the last, 101-th term is 62; the sum of the first 101 terms is


    {{{S[101]}}} = {{{((a[1]+a[101])/2)*101}}} = {{{((-38+62)/2)*101}}} = {{{(24/2)*101}}} = 12*101 = 1212.
</pre>


Now, if you think carefully on this example, you will be able to construct your own examples with common differences 2, 3, and so on.


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On arithmetic progressions, there is a bunch of lessons in this site:

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson>Arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson>The proofs of the formulas for arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson>Problems on arithmetic progressions</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson>Word problems on arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson>Mathematical induction and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson>One characteristic property of arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson>Solved problems on arithmetic progressions</A> 



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


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



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.