Question 1178050
.



            The problem does not define the meaning of S6, S4, so I will suppose that 

            S6 and S4 are the sums of the first 6 and the first 4 terms of the AP, respectively.



<pre>
There are 13 - 6 = 7 gaps between the 13-th and 6-th terms of the AP on the number line,

so each gap is  {{{(77-35)/7}}} = 6 units.


Thus the common difference of the AP is  d= 6.


Now,  {{{S[6]}}} - {{{S[4]}}} = {{{a[5]}}} + {{{a[6]}}} = {{{(a[6]-d)}}} + {{{a[6]}}} = (35-6) + 35 = 70-6 = 64.


It is the answer to the problem's question:  {{{S[6]}}} - {{{S[4]}}} = 64.
</pre>

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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>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Finding-number-of-terms-of-an-arithmeti--progression.lesson>Finding number of terms of an arithmetic progression</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Inserting-arithmetic-means-between-given-numbers.lesson>Inserting arithmetic means between given numbers</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.