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`