<PRE><B>Question 1108752</B>
.
&lt;pre&gt;
{{{S[10]}}} = {{{(a + (9d)/2)*10}}}.    &lt;&lt;&lt;---===  Standard formula for the sum of AP


{{{S[5]}}} = {{{(a + (4d)/2)*5}}}.      &lt;&lt;&lt;---===  Standard formula for the sum of AP


{{{S[10]}}} = {{{4*S[5]}}}  is equivalent to


{{{(a + (9d)/2)*10}}} = {{{4((a + (4d)/2)*5)}}}.


{{{(a + (9d)/2)*2}}} = {{{4*(a+(4d)/2)}}}


2a + 9d = 4a + 8d  ====&gt;  9d - 8d = 4a - 2a  ====&gt;  d = 2a.
&lt;/pre&gt;

&lt;U&gt;Answer&lt;/U&gt;.  &amp;nbsp;&amp;nbsp;d = 2a  &amp;nbsp;is the relation between &amp;nbsp;&quot;d&quot; &amp;nbsp;and &amp;nbsp;&quot;a&quot;.


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There is a bunch of lessons on arithmetic progressions in this site:

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Arithmetic-progressions.lesson&gt;Arithmetic progressions&lt;/A&gt;

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/The-proofs-of-the-formulas-for-arithmetic-progressions.lesson&gt;The proofs of the formulas for arithmetic progressions&lt;/A&gt; 

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Problems-on-arithmetic-progressions.lesson&gt;Problems on arithmetic progressions&lt;/A&gt;  

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Word-problems-on-arithmetic-progressions.lesson&gt;Word problems on arithmetic progressions&lt;/A&gt;

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Mathematical-induction-and-arithmetic-progressions.lesson&gt;Mathematical induction and arithmetic progressions&lt;/A&gt;

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/One-characteristic-property-of-arithmetic-progressions.lesson&gt;One characteristic property of arithmetic progressions&lt;/A&gt;

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=https://www.algebra.com/algebra/homework/Sequences-and-series/Solved-problems-on-arithmetic-progressions.lesson&gt;Solved problems on arithmetic progressions&lt;/A&gt; 



Also, &amp;nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;- &lt;A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson&gt;ALGEBRA-II - YOUR ONLINE TEXTBOOK&lt;/A&gt;.


The referred lessons are the part of this online textbook under the topic &lt;U&gt;&quot;Arithmetic progressions&quot;&lt;/U&gt;.



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.


</PRE>