# Solve 2 Questions: Find Formula for "r" & Write 500050 as Sum of 2 Squares

• Trail_Builder
In summary, the conversation is about two questions regarding compound interest and finding the sum of two square numbers. The first question involves finding a formula for r in terms of n, while the second question involves writing 500050 as the sum of two square numbers. The person asking the questions receives confirmation that their answer for the first question is correct and is given a hint for solving the second question.

#### Trail_Builder

hi, I have 2 questions here that I am stuck on. I thought I may as well stick them in the same topic to avoid cluttering the forum. hope you can help, thnx

QUESTION 1

## Homework Statement

Fred invests an amount of money in an account paying r% compound interest per annum. The amount of money doubles after n years.

Find a formula for r in terms of n.

## The Attempt at a Solution

where m is the initial money]

$$2m = m(\frac{100+r}{100})^{n}$$

$$2 = (\frac{100+r}{100})^{n}$$

$$\sqrt[n]{2} = \frac{100+r}{100}$$

$$100\sqrt[n]{2} = 100 + r$$

$$r = 100\sqrt[n]{2} - 100$$

now, that seems to be the right answer but looks kinda ugly... first of all, is that answer right? and secondly, if so, is there a nicer way to put it :S ?

thnx

QUESTION 2

## Homework Statement

Using this result, or otherwise

I proved for the previous question that

$$(m^2+1)(n^2+1) = (m + n)^2 + (mn -1)^2$$

write 500050 as the sum of 2 square numbers.

## The Attempt at a Solution

I havn't been told how to do this. maybe because this is on a past paper it isn't on the syllabus anymore, but still, I'd like to know how to do it because i doubt its that hard once you know how.

thnx

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correction:

the latex thing won't let me change it despite me trying a billion times. so basically, in the first use of latex the star thing isn't supposed to be there.

if mod or whatever can figure out how to remove the star, please do :D thnx

For 2) write 500050=10001*50 and think about it.

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done it :D

107^2 + 699^2

thnx man

is there actually an elegant way of picking the factors outa 500050 or is it simply guesstimation?

I just picked the obvious factors of 500050. If they had chosen odder ones it would have been much harder as I don't know any really systematic way of finding 'square+1' factors except trial and error.

## What does "r" represent in this formula?

"r" represents the unknown variable that we are trying to solve for in the given equation.

## How do you find the formula for "r"?

To find the formula for "r", we need to use algebraic manipulation and solve for "r" by isolating it on one side of the equation.

## What is the significance of writing 500050 as a sum of 2 squares?

Writing 500050 as a sum of 2 squares is significant because it allows us to represent the number in a more simplified and organized form, making it easier to work with and solve equations involving this number.

## Can you explain the process of finding the sum of 2 squares for 500050?

The process of finding the sum of 2 squares for 500050 involves finding the square root of the number and then expressing it as a sum of 2 squares using the Pythagorean theorem. In this case, the sum of 2 squares for 500050 would be 500^2 + 50^2.

## Is there a specific method for finding the sum of 2 squares for any number?

Yes, there are various methods for finding the sum of 2 squares for any number. One method is to factor the number into its prime factors and then use those factors to express the number as a sum of 2 squares. Another method is to use the formula (a+b)^2 = a^2 + 2ab + b^2 to find the sum of 2 squares for a given number.