Which if the following summation notations represent this series?

Answer:
E.
Step-by-step explanation:
Let's look at the sequence:
[tex]3,8,13,\cdots[/tex]
The sequence has a common difference. We say it has a common difference because 8-3 has the same value as 13-8.
The common difference is what these differences are equal which is 5.
The sequence is therefore linear or arithmetic (same thing).
So it has form [tex]y=mx+b[/tex] where the slope/common difference is 5.
[tex]y=5x+b[/tex]
We need to find the [tex]y-[/tex]intercept [tex]b[/tex].
The list of points is related to the above sequence:
[tex](x,y)[/tex]
[tex](1,3)[/tex]
[tex](2,8)[/tex]
[tex](3,13)[/tex]
I only need one of these points to find [tex]b[/tex].
Let's use the first one since it has smallest numbers and I don't wish to use the calculator unless I really have to.
[tex]y=5x+b[/tex] with [tex](x,y)=(1,3)[/tex]:
[tex]3=5(1)+b[/tex]
[tex]3=5+b[/tex]
Subtract 5 on both sides:
[tex]3-5=b[/tex]
[tex]-2=b[/tex]
So the equation for the set of points is:
[tex]y=5x-2[/tex]
Now they are using [tex]i[/tex] instead of x[/tex]:
[tex]a_i=5i-2[/tex]
Since all summations begin at [tex]i=1[/tex], then the choices are done to either choice C and choice E.
I need to find how many terms are in the given series. The last number is 33. So what term number is associated with the value 33.
[tex]a_i=5i-2[/tex]
[tex]33=5i-2[/tex]
Add 2 on both sides:
[tex]35=5i[/tex]
Divide both sides by 5:
[tex]\frac{35}{5}=i[/tex]
[tex]7=i[/tex]
So there are 7 terms and the 7th term is 33.
So the answer is choice E.