Question

In the 1st generation, there are 3 rabbits in a meadow. Every generation after that, the rabbit population multiplies by 6. In generation 2 there are 18 rabbits, in generation 3 there are 108 rabbits, and so on. Which explicit formula can be used to find the number of rabbits in the nth generation?

Answer 1

A1=3,      a2=3*6,      a3=3*6*6
$a_{n}=a_{1}*6^{n-1} a_{n}=3*6^{n-1} a_{n}= \frac{3*6^{n}}{6} = \frac{6^{n}}{2} a_{n}= \frac{6^{n}}{2}$

Answer 2

Answer:

$a_n=3\times 6^{n-1}\\\\a_n=\frac{6^n}{2}$

Step-by-step explanation:

This is a geometric sequence, since each term is found by multiplying the previous term by a constant (called the common ratio).

The explicit formula for a geometric sequence is

$a_n=a_1\times r^{n-1}$, where a₁ is the first term and r is the common ratio.

In this sequence, the first term is 3 and the common ratio is 6; this gives us

$a_n=3\times 6^{n-1}$

Using the product property of exponents, we can write the power of 6 as

$a_n=3\times 6^n\times 6^{-1}$

Something raised to the -1 power is "flipped"; since 6 = 6/1, this means we have 1/6:

$a_n=3\times 6^n\times \frac{1}{6}$

Using the commutative property, we multiply the 3 and the 1/6:

$a_n=\frac{3}{6}\times 6^n$

Simplifying, we have

$a_n=\frac{6^n}{2}$