Question

Calculus II question on series, just understanding how series are set up. Screenshot will be posted momentarily.

Answer 1

Why is the initial term excluded when finding the nth partial sum of the series?

Answer 2

("This sum is [sum here]..."), why does that exclude the first term of the series?

Answer 3

2ar +2ar^2 +2ar^3+.....
follows a pattern and hence its sum can be found using general formula

but a+2ar +2ar^2 +2ar^3+..... does not follow any pattern

so they found 2ar +2ar^2 +2ar^3+..... first and then added a to it

Answer 4

And then how does that last part (a + (2ar)/(1-r) simplify to a(1+r)/(1-r)? I don't see how they made that jump.