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 nth generation?
Which explicit formula can be used to find the number of monkeys in the nth generation?
Answers
GPT-4o mini
The monkey population doubles every generation, starting with 4 monkeys in the first generation. To find the number of monkeys in the \(n\)th generation, we can express the population as follows:
1. In the 1st generation (when \(n=1\)), the number of monkeys is \(4\).
2. In the 2nd generation (when \(n=2\)), the number of monkeys is \(4 \times 2 = 8\).
3. In the 3rd generation (when \(n=3\)), the number of monkeys is \(4 \times 2^2 = 16\).
4. Continuing this pattern, in the \(n\)th generation, the number of monkeys can be represented as \(4 \times 2^{(n-1)}\).
Thus, the explicit formula for the number of monkeys in the \(n\)th generation is:
\[
M(n) = 4 \times 2^{(n-1)}
\]
Where \(M(n)\) denotes the number of monkeys in the \(n\)th generation.
1. In the 1st generation (when \(n=1\)), the number of monkeys is \(4\).
2. In the 2nd generation (when \(n=2\)), the number of monkeys is \(4 \times 2 = 8\).
3. In the 3rd generation (when \(n=3\)), the number of monkeys is \(4 \times 2^2 = 16\).
4. Continuing this pattern, in the \(n\)th generation, the number of monkeys can be represented as \(4 \times 2^{(n-1)}\).
Thus, the explicit formula for the number of monkeys in the \(n\)th generation is:
\[
M(n) = 4 \times 2^{(n-1)}
\]
Where \(M(n)\) denotes the number of monkeys in the \(n\)th generation.