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Determine the arithmetic sequence if the fourth term is negative six and the eleventh term is negative thirty four

1 Answer

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An arithmetic sequence takes the form


a_n=a_(n-1)+d

where
d is the common difference between terms. You can solve for
a_n in terms of any of the previous terms of the sequence:


a_n=a_(n-1)+d\implies~a_n=a_(n-2)+2d\implies~a_n=a_(n-3)+3d\implies\cdots\implies~a_n=a_(n-k)+kd

for some integer
1\le k\le n-1

Continuing in this way, you know that the sequence can be defined explicitly in terms of the first term
a_1


a_n=a_1+(n-1)d

Given that the 4th term is
a_4=-6 and the 11th term is
a_(11)=-34, you have the following system of equations.


\begin{cases}-6=a_1+(4-1)d\\-34=a_1+(11-1)d\end{cases}

Solving this system for the two unknowns yields
a_1=6 and
d=-4.

So, the sequence is given explicitly by


a_n=6+(n-1)(-4)=-4n+5
User Mick Cullen
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