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.