Geometric sequences and series:
ok so i did a few of them and i am stuck in 1 question here sre my answers for the one i did.
a.) watch the following link, up until time 2: 40. use the appropriate formula to show how much you are paid on the 31st day if you choose the penny a day option.
my 10737418.24
b. use the appropriate formula to show your total earnings for the month if you choose the penny a day option
my 21474836.48

d.) this question is the one i have trouble
suppose you had this job for an infinte amount of time. can you calculate your total earnigs if you are paid with the penny a day option. why or why not.

this is how far i got.. a formula i think should be used is s=a1/1-r

r=2
a1=-0.01
n=31

! it doesnt have to be a full answer just give me tips on how to solve it if you can. you : )

The video link doesn't work for me, but I assume this is the problem involving choosing between two payment types, one that immediately grants you some lump sum of cash, while the other starts off with $0.01 on the first day, $0.02 on the second day, $0.04 on the third, and so on, doubling per day.
Let be the amount of money earned on the -th day. Then is a geometric sequence that satisfies



We can solve explicitly for in terms of the starting payment :







So the amount of money earned on the -th day is



Denote the total amount of money earned over days by . Then we can write



but since we have equivalent expressions for on any given day , this is the same as




Let's call the sum on the right hand side . Notice that







which means we end up with



To answer part (D), you need to look no further than the formula for . The question is basically asking what happens as gets arbitrarily large. It should be clear that grows without bound, so as , the amount of money you would get would diverge to infinity. So technically, you cannot calculate the amount because (1) there's only a finite amount of money to go around, and (2) infinity is not a "computable" number.
If, however, the scale factor used on your income was smaller than 1 - for example, say you were started with a million dollars on the first day, then your income got halved each day - then would eventually converge to 0, and on top of that, would converge to a finite number.