In finance, knowing how to add up a geometric sequence is really important. This skill can help you make smart choices about investments, loans, and savings.

Let's break it down.

A geometric sequence is a list of numbers where each number (after the first) is found by multiplying the previous number by a fixed number, called the common ratio.

For example, if you start with 2 and your common ratio is 3, the sequence will look like this:
2, 6, 18, 54, and so on.

Now, when we speak about the sum of a geometric sequence, we usually want to find the total of the first n terms.

Here’s the formula:

$$S_n = a \frac{(1 - r^n)}{(1 - r)}$$

In this formula,

• S_n is the total of the first n terms,
• a is the first term,
• r is the common ratio, and
• n is how many terms you are adding up.

When the common ratio (r) is greater than 1, this formula can show how investments can grow over time.

In the world of finance, we use the sum of a geometric sequence in important ways:

1. Loan Repayment: When you take out a loan and pay it back in equal payments, the total paid back can be seen as a geometric series. For example, if your loan has interest that adds up, your monthly payments increase because of the extra cost added by the interest.

2. Investment Growth: If you invest a fixed amount of money regularly (like in a savings or retirement account), the total money you save can be seen as a sum of a geometric series. Each time you add more money, it grows at a specific rate due to interest, creating a geometric sequence.

3. Annuities: An annuity pays a certain amount of money regularly. How much these payments are worth now can be calculated using the sum of a geometric sequence. This helps you understand how much you’ll get over time compared to what you put in.

Let’s look at an example to see this in action.

Imagine you invest £1,000 in a bank account that gives you 5% interest each year. If you plan to add £100 at the end of each year for 5 years, you can see this situation as a geometric sequence:

• Initial Investment: £1,000 (This is your first term a)
• Annual Contribution: £100 (Each year, this money grows.)
• Common Ratio: The interest makes this tricky, but each contribution will grow based on when you put it in.

Let’s break down how much money you’ll have after 5 years:

• Year 1: £100 grows for 4 years.
• Year 2: £100 grows for 3 years.
• Year 3: £100 grows for 2 years.
• Year 4: £100 grows for 1 year.
• Year 5: £100 doesn’t grow because it’s just added.

You can use the geometric sum formula to add up these values and see how much you’ll have at the end.

We can also think about a simple example:

Imagine you're thinking of investing in a project that promises to double your money in 5 years. If you invest £10,000:

• Year 0: £10,000
• Year 1: £20,000
• Year 2: £40,000
• Year 3: £80,000
• Year 4: £160,000
• Year 5: £320,000

The money keeps growing, showing how geometric sequences work.

When you calculate the total:

$$S_n = 10,000(1 + 2 + 4 + 8 + 16 + 32) = 10,000(63) = £630,000$$

Understanding how to add these sequences helps you make better decisions about investments and predict future values.

In the same way, when looking at loans, using this geometric series can help you see how payments change what you owe and how much interest you’ll pay overall.

Conclusion:

Knowing how to add a geometric sequence is very important in finance. It helps you figure out future values, understand payments, and make good investment choices. Learning about this gives you a useful tool and helps you become smarter with money.

Whether you’re saving for retirement, assessing an investment, or dealing with a loan, understanding geometric sequences and their sums will help you navigate the tricky world of money management.