Question 1069082
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The 12th term of an arithmetic sequence progression is 5. the 7th term of this progression is 9 more than the 4th term. 
Determine the sum of 20 terms of this progression
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<pre>
"the 7th term of this progression is 9 more than the 4th term." means

{{{a[7]-a[4]}}} = 9.   (1)

But, as you know, {{{a[7]}}} = {{{a[1]+6d}}},  {{{a[4]}}} = {{{a[1]+3d}}}, therefore,

{{{a[7]-a[4]}}} = 6d - 3d = 3d = 9.

Hence,  d = 3.


Now you can determine the first term {{{a[1]}}}  using the condition
"The 12th term of an arithmetic seqence progression is 5.".

It gives you an equation

{{{a[1] + 11*3}}} = 5,

which implies {{{a[1]}}} = 5 - 11*3 = 5 - 33 = -28.


Having {{{a[1]}}} and d, you can calculate the sum of 20 terms of this progression using the formula 

{{{S[20]}}} = {{{(a[1] + ((n-1)*d)/2)*n}}} = {{{(-28 + (19*3)/2)*20}}} = 10.
</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/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, 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>.