a = 2n+1
b = 2n+3
a and b are consecutive odd numbers, where a > b.
Prove that a^{3} - b^{3} is 2 more than a multiple of 24.
Fill in the gaps in the following proof.
a is odd, so a = 2n+1 for some integer n.
Then in terms of n, b =..............
so a^{3} - b^{3} = (2n + 1)^{3} - ...............
Expanding the brackets and simplifying,
a^{3} - b^{3} = ............+ 2which is 2 more than a multiple of 24, as required.
is b = 2n +1 <a ?