Geometric sum - Alfred & interest-rate

• Rectifier
In summary, the problem asks how much money is in Alfred's savings account if he puts 985 USD every year on his birthday, starting at age 35 and continuing until age 48, with an interest rate of 3.7% each year and starting with no money in the account. Using the formula for calculating a geometric sum and considering the balance after each birthday, it can be determined that there will be a total of 985[(((1+i)^n)-1)/i] USD in Alfred's savings account when he is 48 years old.
Rectifier
Gold Member

Homework Statement

Alfred puts 985 USD on his bank account every time he has a birthday. Alfred just turned 48. He started to save money when he turned 35 (including 35th birthday). How much money is there on his savings-account if the interest-rate was 3.7% every year and that he had no money in that account when he started saving.

This problem was translated from Swedish. Sorry for possible grammatical and typographical errors.

Homework Equations

A geometrical sum can be written as:
$$S=\frac{x^{n+1}-1}{x-1}$$

The Attempt at a Solution

This looks lika a geometrical sum. This is the first problem in the chapter and I am already stuck.

Alfreds deposits (not sure if that's the right word) looked like this:
$$985 + 985 \cdot 1.037^1 + ... + 985 \cdot 1.037^m$$
I will explai why i wrote m there soon.

This can be written as:
$$Money = 985(1 +1.037^1 + ... + 1.037^m)$$

The problem I am having is the number that I have to put instead of m. Is it 12, 13 or 14 and what happens when I want to calculate the sum? Will the sum be:

$$Money = 985 \cdot S= 985 \cdot \frac{1.037^{m+1}-1}{1.037-1}$$

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Rectifier said:

Homework Statement

Alfred puts 985 USD on his bank account every time he has a birthday. Alfred just turned 48. He started to save money when he turned 35 (including 35th birthday). How much money is there on his savings-account if the interest-rate was 3.7% every year and that he had no money in that account when he started saving.

This problem was translated from Swedish. Sorry for possible grammatical and typographical errors.

Homework Equations

A geometrical sum can be written as:
$$S=\frac{x^{n+1}-1}{x-1}$$

The Attempt at a Solution

This looks lika a geometrical sum. This is the first problem in the chapter and I am already stuck.

Alfreds deposits (not sure if that's the right word) looked like this:
$$985 + 985 \cdot 1.037^1 + ... + 985 \cdot 1.037^m$$
I will explai why i wrote m there soon.

This can be written as:
$$Money = 985(1 +1.037^1 + ... + 1.037^m)$$

The problem I am having is the number that I have to put instead of m. Is it 12, 13 or 14 and what happens when I want to calculate the sum? Will the sum be:

$$Money = 985 \cdot S= 985 \cdot \frac{1.037^{m+1}-1}{1.037-1}$$

See what works for his 37th or 38th birthday -- something easy like that.

SammyS said:
See what works for his 37th or 38th birthday -- something easy like that.
Lets say 37.

Then

Alfreds deposits would look like this:
$$985 + 985 \cdot 1.037^1 + 985 \cdot 1.037^2$$
37 36 35

This can be written as:
$$Money = 985(1 +1.037^1 + 1.037^2)$$

Does this mean that m=age now - age when he started saving
thus m=13?

Rectifier said:
Lets say 37.

Then

Alfreds deposits would look like this:
$$985 + 985 \cdot 1.037^1 + 985 \cdot 1.037^2$$
37 36 35

This can be written as:
$$Money = 985(1 +1.037^1 + 1.037^2)$$

Does this mean that m=age now - age when he started saving
thus m=13?

You tell us!

There is also the issue of whether you examine the account balance seconds before his birthday, or seconds after it; that will make a difference of 985 in the answer.

Rectifier
Ray Vickson said:
You tell us!

There is also the issue of whether you examine the account balance seconds before his birthday, or seconds after it; that will make a difference of 985 in the answer.
m=age now - age when he started saving

The one above (37)

m=37-35=2

Then this must mean that m in the problem is:
m=48-35=13

Am I right? :D

Rectifier said:
m=age now - age when he started saving

The one above (37)

m=37-35=2

Then this must mean that m in the problem is:
m=48-35=13

Am I right? :D

Make a table. Suppose we look at his balance immediately after his birthday. Then we have:
$$\begin{array}{c|l} & \text{Account}\\ \text{Birthday} & \text{balance} \\ 35 & 985 \\ 36 & 985 + 985 \cdot 1.037 \\ 37 & 985 + 985 \cdot 1.037 + 985 \cdot 1.037^2 \\ \cdots & \cdots \end{array}$$
You can take it from there.

BTW: I recommend you do things in that somewhat detailed way to start with, until you gain a lot of experience with such problems. Jumping right away to plug-in formulas can be a mistake.

Last edited:
Rectifier said:
m=age now - age when he started saving

The one above (37)

m=37-35=2

Then this must mean that m in the problem is:
m=48-35=13

Am I right? :D
Well, it works fine for 37th birthday, so why not for the 48th ?

F= the total money when he is 48
A= every birthday money that is 985
i= %3,7 =0,037
n=48-35=13
F=A(F/A,i,n)
F=985[(((1+i)^n)-1)/i]

is that true?

Last edited:
Thyphon said:
F= the total money when he is 48
A= every birthday money that is 985
i= %3,7 =0,037
n=48-35=13
F=A(F/A,i,n)
F=985[(((1+i^n)-1)/i]

is that true?

the formula is that

Last edited:

1. What is a geometric sum?

A geometric sum is a series of numbers where each term is multiplied by a constant ratio to get the next term. For example, the geometric sum 2 + 4 + 8 + 16 + ... is formed by multiplying each term by 2 to get the next term.

2. Who is Alfred in the context of a geometric sum?

In the context of a geometric sum, Alfred is a fictional character who lends money at a fixed interest rate. He is often used as an example to demonstrate how geometric sums can be used to calculate compound interest over time.

3. How is the interest rate related to a geometric sum?

The interest rate is used as the constant ratio in a geometric sum that represents compound interest. The interest rate is applied to the principal (initial amount) to calculate the interest earned, which is then added to the principal to form the next term of the geometric sum.

4. Can a geometric sum be used to calculate interest for any time period?

Yes, a geometric sum can be used to calculate interest for any time period as long as the interest rate is applied consistently. This means that the interest rate should be expressed in the same time period as the time intervals in the geometric sum. For example, if the interest rate is 5% per year, the time intervals in the geometric sum should also be in years.

5. How can a geometric sum be used in real life?

A geometric sum can be used in real life to calculate compound interest for loans, investments, and savings. It can also be used in other scenarios where a value increases or decreases by a fixed ratio over time, such as population growth or radioactive decay.

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