Question 1200515
.
Find the arithmetic sequence which has the sum of its n terms equal to 2n^2+3n
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<pre>
To solve this problem, you need to know that 


    +----------------------------------------------------+
    |    the sum of any arithmetic progression is        |
    |      a quadratic function of the index "n",        | 
    +----------------------------------------------------+


and vice versa,


    +----------------------------------------------------+
    |   any quadratic function of the integer index "n"  |
    |      creates a unique arithmetic progression.      |
    +----------------------------------------------------+


Based on these facts, it is enough to find the first term of the progression
and the second term, and then calculate its common difference.


    n= 1:  {{{S[1]}}} = 2*1^2+ 3*1 = 5.     {{{a[1]}}} = 5.


    n= 2:  {{{S[1]}}} = 2*2^2+ 3*2 = 14.    {{{a[2]}}} = {{{S[2]}}} - {{{S[1]}}} = 14 - 5 = 9.


This arithmetic progression has the first term  {{{a[1]}}} = 5  and the common difference d= {{{a[2]}}} - {{{a[1]}}} = 9 - 5 = 4.


The progression is  5, 9, 13, 17, . . . 
</pre>

Solved.


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On these properties of arithmetic progression, learn from the lessons

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Free-fall-and-arithmetic-progressions.lesson>Free fall and arithmetic progressions</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Uniformly-accelerated-motions-and-arithmetic-progressions.lesson>Uniformly accelerated motions and arithmetic progressions</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Sequences-and-series/Increments-of-a-quadratic-function-form-an-arithmetic-progression.lesson>Increments of a quadratic function form an arithmetic progression</A>

in this site.