(a) Population Growth Problem

To find the population of a country that increases at a steady rate of 3% per year, starting from 50 million, we use the formula for exponential growth:

$P(t) = P_0 \cdot (1 + r)^t$

where:

• $P(t)$ = population after $t$ years
• $P_0$ = initial population
• $r$ = growth rate (expressed as a decimal)
• $t$ = time in years

In this case:

• $P_0 = 50$ million
• $r = 0.03$ (3%)
• $t = 25$

Substitute the values into the formula:

$P(25) = 50 \cdot (1 + 0.03)^{25}$ $P(25) = 50 \cdot (1.03)^{25}$

Using a calculator, $(1.03)^{25} \approx 2.09377$:

$P(25) = 50 \cdot 2.09377 \approx 104.6885 \text{ million}$

So, the population 25 years later will be approximately 104.69 million.

(b) Compound Interest Problem

We are given a recurrence relation for the amount of money in an account that pays interest at a rate of $i$, compounded $m$ times per year:

$P_k = \left[ 1 + \frac{i}{m} \right] P_{k-1}$

with initial condition $P_0 = P$ (the initial amount deposited).

To find an explicit formula for $P_n$, let’s solve this recurrence relation:

1. Recognize the pattern:

   For $k = 1$: $P_1 = \left[ 1 + \frac{i}{m} \right] P_0$

   For $k = 2$: [ P_2 = \left[ 1 + \frac{i}{m} \right] P_1 = \left[ 1 + \frac{i}{m} \right]^2 P_0 ]

   For $k = 3$: [ P_3 = \left[ 1 + \frac{i}{m} \right] P_2 = \left[ 1 + \frac{i}{m} \right]^3 P_0 ]

   Continuing this pattern, for any $n$: $P_n = \left[ 1 + \frac{i}{m} \right]^n P_0$

2. Final formula:

   $P_n = P \cdot \left[ 1 + \frac{i}{m} \right]^n$

This formula provides the amount of money after $n$ compounding periods, where $P$ is the initial deposit, $i$ is the annual interest rate, and $m$ is the number of times the interest is compounded per year.

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Do you want more details on these explanations or have any questions?

5 Related Questions:

1. How does the compound interest formula change if the compounding is continuous?
2. What would the population be after 50 years if the growth rate remains the same?
3. How can you derive the formula for exponential growth from differential equations?
4. If a bank offers different compounding frequencies (e.g., monthly, quarterly), how does that affect the final amount?
5. What is the difference between simple interest and compound interest calculations?

Tip:

Always convert percentage rates into decimals when using them in formulas to avoid errors in calculations. For example, 3% becomes 0.03 when substituted into formulas.