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A population grows according to an exponential growth model, with [tex]$P_0=60$[/tex] and [tex]$P_1=102$[/tex].

Complete the recursive formula:
[tex]P_n = \square \times P_{n-1}[/tex]

Write an explicit formula for [tex]P_n[/tex]:
[tex]P_n = 60 \times (\square)^n[/tex]


Sagot :

Certainly! Let's break down the solution step-by-step for the given problem where the population grows according to an exponential growth model, with [tex]\( P_0 = 60 \)[/tex] and [tex]\( P_1 = 102 \)[/tex].

### Step 1: Determine the Multiplication Factor

To find the multiplication factor used in the recursive formula, we use the initial values [tex]\( P_0 \)[/tex] and [tex]\( P_1 \)[/tex]:
[tex]\[ P_0 = 60 \][/tex]
[tex]\[ P_1 = 102 \][/tex]

The multiplication factor is calculated as the ratio of [tex]\( P_1 \)[/tex] to [tex]\( P_0 \)[/tex]:
[tex]\[ \text{Multiplication Factor} = \frac{P_1}{P_0} = \frac{102}{60} = 1.7 \][/tex]

### Step 2: Write the Recursive Formula

Using the multiplication factor, the recursive formula can be written as:
[tex]\[ P_n = 1.7 \times P_{n-1} \][/tex]

### Step 3: Determine the Explicit Formula

The explicit formula for [tex]\( P_n \)[/tex] in an exponential growth model can be written using the initial population [tex]\( P_0 \)[/tex] and the multiplication factor. The formula is:
[tex]\[ P_n = P_0 \times (\text{Multiplication Factor})^n \][/tex]

Given [tex]\( P_0 = 60 \)[/tex] and the multiplication factor is [tex]\( 1.7 \)[/tex], the explicit formula becomes:
[tex]\[ P_n = 60 \times (1.7)^n \][/tex]

### Final Answer

Recursive Formula:
[tex]\[ P_n = 1.7 \times P_{n-1} \][/tex]

Explicit Formula:
[tex]\[ P_n = 60 \times (1.7)^n \][/tex]
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