Question

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

Answer 1

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

$a_n=6+(n-1)(-4)=-4n+5$