Question involving summing a geometric series. (1 Viewer)

[nicholas101]

I am currently going through the series and sequence topic in HSC right now and I am struggling with questions from time to time, but this one is tricky:
"The number of people attending the yearly Abletown Show is rising by 5% per annum and the number attending the yearly Bush Creek Show is falling by 5% per annum. In the first year under consideration, 5000 people attended both shows.
a) Find the total number attending each show during the first six years.
b) Show that the number attending the Abletown Show first exceeds ten times the number attending the Bush Creek Show in the 25th years.
c) What is the ratio (correct to three significant figures) of the total number attending the Abletown Show over these 25 years to the total attending the Bush Creek Show?"
I've finished a and b, so I need help with question c. I have asked my friends and teacher, but they aren't available unfortunately, so help would be appreciated.

Thank you!

 

[Pedro123]

I won't do the question for you, but from what I understand, you need to sum up the geometric series to that point. The first 2 only ask you to find certain terms, and for the next one you need to sum those terms. As such, you use the formula:

Where a is the original starting term, r is the rate of change and n is the number of periods that the rate r is applied for. You do this for both schools, then find the ratio between them.
Another interesting thing is the limiting sum. So say you wanted to sum 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...
That would be equivalent to

where the variable mean the same thing (Plugging those values show that the sum approaches 2