Doing Mathematics: Playing With a Simple Pattern
2 + 2 = 2 x 2
The above statement is very obvious if one understands how addition and multiplication work conceptually (or even visually).
But, what if you push this statement a little bit. Is there a number n that you can add to 3 and multiply by 3 to get the same result?
Now, continue exploring this question and others such as:3 + n = 3 * n
4 + n = 4 * n
5 + n = 5 * n
6 + n = 6 * n
7 + n = 7 * n
Does a pattern develop? Can you justify it using algebra?
Finally, what happens if you consider the complementary pattern of 2 - n = 2 / n...and again, try numbers other than 2?
Hint: You should see a pattern...for 2 we see n is 2, for 3 we see n is 3/2, for 4 we see n is 4/3...
Solution Commentary: Can you justify the observed pattern:
n + n/(n-1) = n * n/(n-1)