Question 1119318
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<pre>
The first term is 112 = 14*8.


The sum is  112 + 126 + 140 + . . .  .


To find the last term and the number of terms, calculate the number of intervals of the length 14 starting from 112 and ending 2000.


The number of such intervals is the INTEGER PART of the number  {{{(2000-112)/14}}} = 134.86, i.e. 134.


Thus the number of terms multiple to 14 from 100 (or 112) to 2000 is 135.


Check:  112 + 134*14 = 1988.


Now use the formula for the sum of the first n terms of an arithmetic progression  


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


In your case  {{{a[1]}}} = 112;  n = 135;  {{{a[n]}}} = 1988.


Hence, the sum  S = {{{((112+1988)/2)*135}}} = 141750.


It is your answer.
</pre>

Solved.


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For introductory lessons on arithmetic progressions see 

&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>

in this site.


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.