Question 1098032
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<U>The MENTAL solution by the "guessing method"</U>


<pre>
{{{a[1]}}} = 1;

{{{a[2]}}} = 3;

d = 2 (the common difference);


<U>Check</U>:  {{{a[10]}}} = {{{a[1]}}} + d*(10-1) = 1 + 2*9 = 19  ! Correct !


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{{{a[5]}}} = 1 + 4*2 = 9;

{{{a[6]}}} = 1 + 5*2 = 11;

{{{a[5]}}} + {{{a[6]}}} = 9 + 11 = 20.
</pre>

Solved.



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Surely, it can be solved algebraically 


<pre>
{{{a[1]}}} + {{{(a[1]+d)}}} = 4,      (1)
{{{a[10]}}} = {{{a[1] + 9*d}}} = 19,   (2)

which gives you the system of two equations in two unknowns

{{{2a[1] + d}}} = 4,
{{{a[1]+9d}}} = 19,


You can solve it by any method you know/you want to get the same answer.
</pre>


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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=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.