Question

If you invest \(£ 1000\) on the first day of each year, and interest is paid at \(5 \%\) on your balance at the end of each year, how much money do you have after 25 years?

Short answer

After 25 years, you will have approximately £47720.

Step-by-step solution

Identify the Problem

Determine how much money is accumulated after 25 years if £1000 is invested at the beginning of each year with a 5% annual interest rate compounded yearly.

Understand Compound Interest Formula

For compound interest, the formula used to calculate the future value is \[ A = P \left( 1 + \frac{r}{n} \right)^{nt} \]where \(A\) is the amount of money accumulated after \(n\) years, including interest, \(P\) is the principal amount (initial investment), \(r\) is the annual interest rate (decimal), \(n\) is the number of times interest is compounded per year, and \(t\) is the number of years the money is invested.

Set Variables for the First Investment

For the first investment:- \(P = £1000\)- \(r = 0.05\)- \(n = 1\)- \(t = 25\)

Calculate the Future Value of the First Investment

Using the variables for the first investment:\[ A_1 = 1000 \left( 1 + \frac{0.05}{1} \right)^{1 \times 25} = 1000 \left( 1.05 \right)^{25} \approx 3386.35 \]

Repeat for Each Subsequent Year

For the second investment (made at the beginning of the second year) and subsequent investments, adjust the number of years accordingly:- For the second investment: \(t = 24\)- For the third investment: \(t = 23\)... and so forth until the final investment

Summing All Future Values

Each year's investment will grow to different amounts depending on the number of years it accrues interest. Sum these amounts:\[ Total = 1000(1.05)^{25} + 1000(1.05)^{24} + ... + 1000(1.05)^{1} + 1000 \]

Use Geometric Series Formula

The total sum can be expressed as a geometric series:\[ S = 1000 \sum_{k=0}^{24} (1.05)^k \]Using the geometric series sum formula for \( S_n = a \frac{r^n - 1}{r - 1} \):\[ S = 1000 \frac{(1.05)^{25} - 1}{1.05 - 1} \approx 1000 \frac{(3.386 - 1)}{0.05} \approx 1000 \times 47.72 = 47720 \]

Conclusion

The total amount accumulated over 25 years, after investing £1000 at the beginning of each year with 5% annual interest, is approximately £47720.

Key concepts

future value

The future value is the total amount of money accumulated after an investment, including interest. It helps to predict how much your investment will be worth after a specified period.
If you invest a principal amount today, the future value will indicate what that principal, plus earned interest, will amount to at the end of the investment period.
In our exercise, we discover the future value by considering yearly investments and the interest rate.

• Each year, a £1000 investment retains interest, building up over time.
• For example, £1000 invested for 25 years at 5% interest grows to about £3386.35.

By calculating the future values for all yearly investments and summing them up, we get the total future value after 25 years, which is about £47720.

principal amount

The principal amount is the initial sum of money invested or loaned. It's key for calculating interest and future value.
In our example, this is the £1000 invested at the beginning of each year.

• The principal is consistent every year (£1000 each time).
• Understanding the principal is crucial, as it forms the base on which interest is calculated.

In our compound interest formula, 'P' represents the principal. With a consistent principal, we can simplify calculations using sums of geometric sequences to determine the total future value.

geometric series

A geometric series sums terms that have a constant ratio between successive terms. It's particularly useful for understanding compound interest.
Each year we invest £1000, and each £1000 grows with compound interest at 5%. This forms a geometric series.

• For each subsequent year, the number of years the money is invested decreases (starting from 25 years down to 1 year).
• The series can be summed using the formula: \[ S = 1000 \frac{(1.05)^{25} - 1}{1.05 - 1} \]

This formula helps us calculate the total future value by simplifying the sum of all individual yearly investments and their compounded interests. Summing up, we determine that the total amount accumulated over 25 years is approximately £47720.