Question is from Fibonacci\'s Liber Quadratorum
Given the squares of three successive odd numbers, show that thelargest square exceeds the
middle square by eight more than themiddle square exceeds the smallest.
I think I have the answer (2n+5)2 = (2n+3)2 +[(2n+3)2 - (2n+1)2] + 8 but I am not sure howto
show it algebraically.
Solution
three consecutive odd numbers: (2x+1), (2x+3), (2x +5) (2x+1)2 = 4x2 + 4x + 1
(2x+3)2 = 4x2 + 12x + 9 (2x+5)2 = 4x2 + 20x + 25 largest square - middle square = 4x2 + 20x
+ 25 -(4x2 + 12x + 9) = 8x + 16 middle square - smallest square = 4x2 + 12x + 9 -(4x2 + 4x + 1)
= 8x + 8 so the difference between the above two is 8x+16 - (8x+8) = 8.
Question is Give a definition of each of the following, and then st.pdf
1. Question is: Give a definition of each of the following, and then state a characterization of each.
(a) two lines in the plane are perpendicular
(b) a rational number
Thanks!
Solution
a angle between the both the lines is 90degrees b number that cannot be written in
p/q form
