Question

In the 1st generation, there are 4 monkeys on an island. Every generation after that, the monkey population doubles. In generation 2, there are 8 monkeys, in generation 3 there are 16 monkeys and so on.

Which explicit formula can be used to find the number of monkeys in the (n)th generation?

a(n) = 4•(1/2)^(n-1)
a(n) = 2•(1/4)^(n-1)
a(n) = 4•2^(n-1)
a(n) = 2•4^(n-1)

Answer 1

To find the explicit formula for the number of monkeys in the \( n \)th generation, we can observe the pattern of the population growth.

We start with 4 monkeys in the 1st generation:

- Generation 1: \( 4 \) monkeys
- Generation 2: \( 4 \times 2 = 8 \) monkeys
- Generation 3: \( 8 \times 2 = 16 \) monkeys
- Generation 4: \( 16 \times 2 = 32 \) monkeys

We can see that the number of monkeys doubles every generation. This means that the population in the \( n \)th generation can be expressed as:

\[
a(n) = 4 \times 2^{(n-1)}
\]

Starting with 4 monkeys, and for each generation (n-1) we multiply by 2.

Thus, the correct explicit formula that can be used to find the number of monkeys in the \( n \)th generation is:

\[
\boxed{a(n) = 4 \cdot 2^{(n-1)}}
\]

So the correct choice from your options is:

**a(n) = 4•2^(n-1)**.