A confusing question on sequences and series, please help!

Question

A bank is undergoing financial crisis and every year their starting budget is half the value of the previous year. Supposing that the bank started with $1 000 000 in the first year and they spend all its starting budget by the end of every year, how much money would it have spent before it goes bankrupt (i.e when the annual starting budget becomes $0)?

Answer 1

This is an infinite geometric series.

year 1: 1,000,000

year 2: 500,000

year 3: 250,000

year 4: 125,000

etc.

The budget reduces by half every year but technically never reaches zero. However, the sum can still be calculated.

S = 1000000 / (1 - 1/2)

= $2,000,000

It may be reasonable to assume that once the annual budget drops below $0.005, it will be rounded to zero and at such time the company will be bankrupt. So, after about 28.6 years.

1000000(1-.5)^(n-1) = 0.005

n ≈ 28.6 years