Question 1068258: Discrete mathematics.
A colony of bats is counted every 2 months. The first four counts are 1200, 1800, 2700, and 4050.
a. Assuming that this growth rate continues, write a recurrence relation for the number of bats at count n.
b. Solve the recurrence relation.
c. What will the 12th count be?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! start with 1200
next is 1800 which is 1.5 * 1200.
next is 2700 which is 1.5 * 1800.
next is 4050 which is 1.5 * 2700.
it looks like the population of bats is increasing 1.5 times every 2 months.
the formula would be b.n = b.0 * 1.5 ^ n
b.n is the future population.
b.0 is the present population of bats which is 1200.
1.5 is the growth factor every 2 months.
n is the number of 2 month periods.
using this formula, you get the following:
after 1 two month period, the population is 1200 * 1.5 = 1800.
after 2 two month periods, the population is 1200 * 1.5 ^ 2 = 2700.
after 3 two month periods, the population is 1200 * 1.5 ^ 3 = 4050.
after 12 two month periods, the population is 1200 * 1.5 ^ 12 = 155,695.6055.
this is not the recurrence formula, or what might be called a recursive formula, however.
the recurrence formula would be, to the best of my knowledge, b.(n+1) = b.n * 1.5, where b.0 is equal to 1200.
so you start with b.0 = 1200
b.(n+1) = b.n * 1.5 becomes b.1 = b.0 * 1.5 which gets you b.1 = 1200 * 1.5 = 1800
b.(n+1) = b.n * 1.5 then becomes b.2 = b.1 * 1.5 which gets you b.2 = 1800 * 1.5 = 2700.
each succeeding time period results in a bat population which is 1.5 times the previous time period.
the succession becomes:
b.0 = 1200
b.1 = 1800
b.2 = 2700
b.3 = 4050
b.4 = 6075
b.5 = 9112.5
b.6 = 13,668.75
b.7 = 20,503.125
b.8 = 30,754.6875
b.90 = 46,132.03125
b.10 = 69,198.04688
b.11 = 103,797.0703
b.12 = 155,695.6055
note that b.n means b sub n which means the nth occurrence of the 2 month growth periods.
you start with b.0 which is equal to 1200.
next is b.1 which is 1.5 * 1200 which is equal to 1800.
i used the b to indicate the number of bats in the nth growth period.
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