Question

The growth of a local raccoon population approximates a geometric sequence where an is the number of raccoons in a given year and n is the year. after 6 years there are 45 raccoons and after 8 years there are 71 raccoons.

Answer 1

Answer:

 GENERAL EXPLICIT SEQUENCE IS GIVEN   $a_n = (14.74)(r)^{n-1)}$

Step-by-step explanation:

Let n be the number of year the data is recorded in.

a: The number of raccoons taken initially.

r: The multiplying factor

$a_n$  : The number of raccoon in the nth year.

As given:  $a_6 = 45, a_8 = 71$

Now, as the given situation can be expressed as GEOMETRIC SERIES:

$a_n = a r^{(n-1)}$

Applying the same to given terms, we get:

$a_6 = a r^{(6-1)} = ar^5 = 45\\\implies ar^5 = 45$

$a_8 = a r^{(8-1)} = ar^7 =71\\\implies ar^7 = 71$

Dividing both equations, we get:

$\frac{ar^7}{ar^5} = \frac{71}{45} \\\implies r^2 = 1.58\\\implies r = 1.25$

So, the first term $a = \frac{45}{(1.25)^5} = 14 .74 \approx 15$

So, the GENERAL EXPLICIT SEQUENCE IS GIVEN as:  $a_n = (14.74)(r)^{n-1)}$