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Consider a population that grows according to the recursive rule [tex]\( P_n = P_{n-1} + 95 \)[/tex], with initial population [tex]\( P_0 = 80 \)[/tex].

Then:
[tex]\[
\begin{array}{l}
P_1 = \square \\
P_2 = \square
\end{array}
\][/tex]

Find an explicit formula for the population. Your formula should involve [tex]\( n \)[/tex].

[tex]\[
P_n = \square
\][/tex]

Use your explicit formula to find [tex]\( P_{100} \)[/tex].

[tex]\[
P_{100} = \square
\][/tex]

Sagot :

GinnyAnswer
To solve the given population growth problem step-by-step, let's follow the outlined tasks methodically:

1. Calculate [tex]\( P_1 \)[/tex]:

Given the initial population [tex]\( P_0 = 80 \)[/tex] and the recursive rule [tex]\( P_n = P_{n-1} + 95 \)[/tex]:

[tex]\[ P_1 = P_0 + 95 = 80 + 95 = 175 \][/tex]

So, [tex]\( P_1 = 175 \)[/tex].

2. Calculate [tex]\( P_2 \)[/tex]:

Using the previously found [tex]\( P_1 \)[/tex] and the same recursive rule:

[tex]\[ P_2 = P_1 + 95 = 175 + 95 = 270 \][/tex]

So, [tex]\( P_2 = 270 \)[/tex].

3. Find an explicit formula for the population [tex]\( P_n \)[/tex]:

The recursive rule [tex]\( P_n = P_{n-1} + 95 \)[/tex] suggests that each subsequent term increases by 95. To derive an explicit formula, observe that each step adds an additional 95. Starting from [tex]\( P_0 \)[/tex]:

- [tex]\( P_1 = P_0 + 95 \)[/tex]
- [tex]\( P_2 = P_0 + 2 \times 95 \)[/tex]
- [tex]\( P_3 = P_0 + 3 \times 95 \)[/tex]

Generalizing this:

[tex]\[ P_n = P_0 + n \times 95 \][/tex]

Substituting [tex]\( P_0 = 80 \)[/tex]:

[tex]\[ P_n = 80 + n \times 95 \][/tex]

So, the explicit formula is [tex]\( P_n = 80 + n \times 95 \)[/tex].

4. Calculate [tex]\( P_{100} \)[/tex] using the explicit formula:

Now, substitute [tex]\( n = 100 \)[/tex] into the formula [tex]\( P_n = 80 + n \times 95 \)[/tex]:

[tex]\[ P_{100} = 80 + 100 \times 95 = 80 + 9500 = 9580 \][/tex]

So, [tex]\( P_{100} = 9580 \)[/tex].

Summary:

[tex]\[ P_1 = 175 \][/tex]

[tex]\[ P_2 = 270 \][/tex]

[tex]\[ P_n = 80 + n \times 95 \][/tex]

[tex]\[ P_{100} = 9580 \][/tex]
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