# Geometric sum - Alfred & interest-rate

1. Jun 7, 2015

### Rectifier

1. The problem statement, all variables and given/known data
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.

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

3. 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 thats 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:

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

Last edited: Jun 7, 2015
2. Jun 7, 2015

### SammyS

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

3. Jun 7, 2015

### Rectifier

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?

4. Jun 7, 2015

### Ray Vickson

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.

5. Jun 7, 2015

### Rectifier

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

6. Jun 7, 2015

### Ray Vickson

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: Jun 8, 2015
7. Jun 7, 2015

### SammyS

Staff Emeritus
Well, it works fine for 37th birthday, so why not for the 48th ?

8. Jun 30, 2015

### Thyphon

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: Jun 30, 2015
9. Jun 30, 2015

### Thyphon

the formula is that

Last edited: Jun 30, 2015