Question 1090952
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<pre>
Use the formula for the sum of the first 50 terms of an arithmetic progression with the common difference 7,

{{{(a[1]+a[50])/2}}}.{{{50}}} = 12075.     (1)


Since {{{a[50]}}} = {{{a[1]+7*49}}}, you can rewrite (1) in the form


{{{(a[1] + a[1]+7*49)/2}}}.{{{50}}} = 12075,   or

{{{(2a[1] + 7*49)}}}.{{{50}}} = 24150,

{{{2a[1]}}} = {{{24150/50 - 7*49}}}  ====>  {{{a[1]}}} = {{{(483 - 7*49)/2}}} = 70.


Thus you found the smallest term of the AP. It is 70.


Then the biggest term is 70 + 7*49 = 413.


Happily, the first term is multiple of 7 and the common difference is 7, so all the terms of the progression 
are consecutive integers multiple of 7.
</pre>

Solved.



There is a bunch of lessons on arithmetic progressions 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>.