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.
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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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