Cubes and the differences of squares Let us investigate the following hypothesis: "Every number cubed can be written as the difference of tow squares" For example: 3^3 = 6^2-3^2 (as 3^3 = 27 and 6^2 - 3^2 = 36 -9 =27) 4^3 = 10^2 - 6^2 (as 4^3 = 64 and 10^2 - 6^2 = 100 - 36 =64) In fact, the hypothesis is true. Given any positive integer n greaterthanorequalto 2, we can find two triangular numbers, a and b, such that n^3 = a^2 - b^2. The first six triangular numbers are shown below. a) investigate by finding the missing triangular numbers for the following cubes. Show working to prove the statement is correct for these instances. 2^3 = ()^2 - ()^2 5^3 = ()^2 - ()^2 The nth triangular number is given by the formula, n(n+1)/2 b) What is the formula for the previous triangular number, i.e., the triangular number before the nth one? c) Show, by providing working that (nth triangular number)^2 - [(n - l)th triangular number]^2 = n^3