SOLUTION: In an arithmetic progression, the thirteenth term Is 27, and the seven term is three times the second term.find the first term and the common differences.

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Question 1036737: In an arithmetic progression, the thirteenth term
Is 27, and the seven term is three times the second term.find the first term and the common differences.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
In an arithmetic progression, the thirteenth term
is 27, and the seventh term is three times the second term.
Find the first term and the common differences.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

We are given 

 = 27,         (1)
  = .      (2)

which, in other terms, is

 + 12*d = 27,           (1')
 +  6*d = .   (2').

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

   + 12*d = 27,         (1'')
 +  3*d =  0.         (2'')

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

 = 27,   or    = 27,  which gives you   = 3.

Then from (2'')  3d =  = 6.  Hence,  d = 2.

Answer.   = 3,  d = 2.


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