Question

A population grows according to an exponential growth model, with Po = 50 and P1 = 70 Complete the recursive formula: Pn - XPR-1 Write an explicit formula for Pn PR

Answer 1

To complete the recursive formula for the population growth model, we can use the given values for P0 and P1 to solve for the unknown variables X and R. Using the formula Pn = (X * Pn-1) + R, we substitute the values and solve the resulting system of equations to find that X = 0.4 and R = 50. Thus, the completed recursive formula is Pn = (0.4 * Pn-1) + 50, and the explicit formula is Pn = 50(1 + 0.4)n.

Step-by-step explanation:

To complete the recursive formula, Pn = (X * Pn-1) + R, we need to determine the values of X and R.

Given that P0 = 50 and P1 = 70, we can substitute these values into the formula:

P1 = (X * P0) + R

Since P0 = 50 and P1 = 70, we have:

70 = (X * 50) + R

To find the values of X and R, we need one more equation. We can use the fact that the population grows exponentially, which means that the growth rate is constant:

Pn = Pn-1 + XPRn-1

= Pn-1(1 + XR)

Since the growth rate is constant, we can rewrite this equation as:

Pn = Pn-1(1 + X)

Using the given values P0 = 50 and P1 = 70, we can substitute these values into the equation:

70 = 50(1 + X)

Solving this equation, we find that X = 0.4.

Now that we know the value of X, we can substitute it back into the equation 70 = (X * 50) + R to find the value of R:

70 = (0.4 * 50) + R

Solving this equation, we find that R = 50.

Therefore, the completed recursive formula is:

Pn = (0.4 * Pn-1) + 50.

The explicit formula for Pn is:

Pn = P0(1 + X)n = 50(1 + 0.4)n.