First we write out our given information:
$$a_8=2a_2$$
$$a_{11}=18$$
$a_n$ is an arithmetic sequence.
Where here $a_n$ means the $n$th term of our sequence.
What does an arithmetic sequence mean? It means to get to the next term in your sequence you add a constant ($c$) each time:
$$a_{n+1}=a_n+c$$
Equivalently:
$$\frac{a_{n+1}-a_{n}}{(n+1)-n}=c$$
So $a_n$ is of slope $c$ ($c_2$ is another constant):
$$a_n=cn+c_2$$
Where here $c_2=a_0$ (Substitute in $n=0$ and see why that has to be the case if we let $a_0$ exist)
Now we use the other given information to try to come up with a solution.
Let $n=2$:
$$a_2=2c+c_2 {}{}$$
Let $n=8$, using the above equation we have:
$$a_8=8c+c_2=2a_2=4c+2c_2 {}{}{}{}$$
Let $n=11$
$$a_{11}=18$$
$$a_{11}=11c+c_2$$
But $a_{11}-a_8=(11c+c_2)-(8c+c_2)=3c$
Hence, $a_{11}=3c+a_{8}$
$$a_{11}=3c+4c+2c_2=18$$
$$a_{11}=3c+8c+c_2=18$$
Solve this system of equations to get a closed form for the arithmetic sequence.
$$a_n=1.2n+4.8$$.
We can check it works $a_2=1.2(2)+4.8=7.2$. Now we compute $a_8$ to see if $a_8=2a_2$ as required: $a_8=1.2(8)+4.8=14.4=2(7.2)=2a_2$. It is arithmetic as we may check $a_{n+1}-a_n$ is a constant $1.2$. Also $a_{11}=1.2(11)+4.8=18$ as required.
The answers follow from this, from summation formulas/methods of evaluating sums, and from algebra.
Best Answer
Hints:
The sum of the first four terms is $a+(a+d)+(a+2d)+(a+3d)=4a+6d$.
The fourth term is $a+3d$.
The eighteenth term is $a+17d$.