Question:
Can you please help me with a general formula for the following system:
2+2=2*2
3+3/2=3*3/2
4+4/3=4*4/3
and how will you prove the formula?
Hi Tor,
This question illustrates the usefulness of the algebraic idea of using letters to stand for numbers. If you rewrite the first equation as
2+2/1=2*2/1
you can see the pattern (equation) and put it into words.
"The sum of any positive integer greater than one with the fraction formed by this number divided by the number which is one less than this number is equal to the product formed by multiplying this number by the fraction formed by this number divided by the number which is one less than this number."
It is certainly easier to write this equation using symbols.
Let n be an integer, greater than 1. Can you prove the following equation is valid?
n + n/(n - 1) = n
n/(n - 1)
The right side can be written (n n)/(n - 1) but for the left side you need a common denominator
n + n/(n - 1)
= n(n - 1)/(n - 1) + n/(n - 1)
= (n(n - 1) + n)/(n - 1)
= (n
= (n
Hence the left side and right side of
n + n/(n - 1) = n
are each equal to (n n)/(n - 1) and thus the equation
n + n/(n - 1) = n
is valid.
Penny