Question 1036737
.
In an arithmetic progression, the thirteenth term  
is 27, and the {{{highlight(cross(seven))}}} seventh term is three times the second term. 
Find the first term and the common differences.
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We are given 

{{{a[13]}}} = 27,         (1)
{{{a[7]}}}  = {{{3*a[2]}}}.      (2)

which, in other terms, is

{{{a[1]}}} + 12*d = 27,           (1')
{{{a[1]}}} +  6*d = {{{3*(a[1]+d)}}}.   (2').

Simplify it and write in the canonical form for the system of two equations in two unknowns:

  {{{a[1]}}} + 12*d = 27,         (1'')
{{{-2a[1]}}} +  3*d =  0.         (2'')

Now, based on (2''), replace the term 12*d in (1'') by {{{8a[1]}}}. You will get a single equation for {{{a[1]}}}

{{{a[1] + 8a[1]}}} = 27,   or   {{{9*a[1]}}} = 27,  which gives you  {{{a[1]}}} = 3.

Then from (2'')  3d = {{{2a[1]}}} = 6.  Hence,  d = 2.

<U>Answer</U>.  {{{a[1]}}} = 3,  d = 2.
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