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Let's solve the problem step by step.

We are given:
- Initial deposit: K1000
- Annual interest rate: 10% (or 0.10)
- Interest is compounded annually.
- \( P_n \) represents the amount in the account after \( n \) years.

### Part (a) – The Recurrence Relation:
The recurrence relation describes how the amount in the account changes from one year to the next.

Since the interest is compounded annually at 10%, the amount in the account after one year is the amount from the previous year plus 10% of that amount. 

Mathematically:
\[
P_{n+1} = P_n \times (1 + 0.10)
\]
or simplified:
\[
P_{n+1} = 1.1 \times P_n
\]

### Part (b) – The Initial Condition:
The initial condition is the starting amount of money in the account, which is K1000. So:
\[
P_0 = 1000
\]

### Part (c) – The Solution to the Recurrence Relation:
To solve the recurrence relation \( P_{n+1} = 1.1 \times P_n \), we notice that this is a geometric progression. The general solution for a recurrence of this type is:

\[
P_n = P_0 \times (1.1)^n
\]

Substituting \( P_0 = 1000 \):
\[
P_n = 1000 \times (1.1)^n
\]

### Part (d) – How Much Will Be in the Account After 50 Years?
To find the amount after 50 years, substitute \( n = 50 \) into the formula:

\[
P_{50} = 1000 \times (1.1)^{50}
\]

Now, let's calculate this.

After 50 years, the amount in the account will be approximately K117,390.85.

Would you like more details or have any questions?

Here are five related questions for further exploration:
1. What if the interest were compounded semi-annually instead of annually?
2. How would the total change if the interest rate were 8% instead of 10%?
3. How long would it take for the account to double in value at a 10% interest rate?
4. What is the formula for calculating the future value of a deposit with continuous compounding?
5. How does the concept of compound interest relate to exponential growth in mathematics?

**Tip:** Understanding recurrence relations helps in solving real-world problems involving interest, population growth, and other cumulative processes.

Suppose that a person deposited K1000 in a savings account at a bank yielding 10% per year with interest compounded annually. Let P_n denote the amount in the account after n years. Find (a) the recurrence relation, (b) the initial condition, (c) the solution to the recurrence relation, and (d) how much will be in the account after 50 years?

Solve a compound interest problem involving a deposit of K1000 with a 10% annual interest rate. Learn how to find the recurrence relation, initial condition, and amount in the account after 50 years. This problem covers geometric progression and exponential growth.

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