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

[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.

Homework Equations

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.

Homework Equations

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

 

[Trail_Builder]

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

 

[Dick]

Your answer to question 1) is correct.

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

 

[Trail_Builder]

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?

 

[Dick]

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.