Answer
The first term is $a_1=-11$
The nth term of an arithmetic sequence is: $a_n=-11+2(n-1) \quad \text{or} \quad a_n=-13+2n$.
Work Step by Step
The $n^{th}$ term of an arithmetic sequence can be found as: $a_n=a_1+(n-1)d$.
where, $a_1$ is the first term and $d$ is the common difference(is the difference between a term and the term proceeding it), that is, $d=a_n-a_{n-1}$
From the given sequence, we have:
$$a_5=-3\quad \text{and}\quad a_{15}=17$$
Note that the $15^{th}$ term can be found by adding the common difference $d$ 10 times to the $5^{th}$ term. Thus,
$$a_{15}=a_5+10 \cdot d$$
Plug $17$ for $a_{15}$ and $-3$ for $a_5$ in the above formula to compute the common difference $d$.
$$a_{15}=a_5+10d \\ 17=-3+10d \\
20=10d\\
\dfrac{20}{10}=d\\
2=d$$
Thus, the $n^{th}$ term of the given sequence is given by the formula
$$a_n=a_1+2(n-1)$$
Use $a_5=-3$ in the formula to obtain:
$a_5=a_1+2(5-1)\\
-3=a_1+2(4) \\
-3=a_1+8\\
-3-8=a_1\\
-11=a_1$
Therefore, the nth term of an arithmetic sequence is:
$$a_n=a_1+2(n-1)\\
a_n=-11+2(n-1)$$
This can be expanded and simplified to:
$$a_n=-11+2n-2\\
a_n=-13+2n$$
