Victor the vampire lived for 200 years in his first life. Each life after that was only one- half as long as the previous life. How many lives did Victor live that were more than one year long?
Victor's second life was 100 years long, his third life was 50 years long, his fourth life was 25 years long, and so on. We can write this as a geometric sequence with first term 200 and common ratio 1/2:
200, 100, 50, 25, ...
To determine the number of lives that were more than one year long, we need to find how many terms of this sequence are greater than 1. We can write this inequality for the nth term of the sequence:
(1/2)^(n-1) * 200 > 1
Simplifying, we get:
(1/2)^(n-1) > 1/200
Taking the logarithm of both sides (with base 1/2), we get:
n - 1 < log(1/200) / log(1/2)
n - 1 < 8.64385619
n < 9.64385619
Since n must be a positive integer, the largest possible value for n is 9. Therefore, Victor lived 9 lives that were more than one year long.