F.4 M2 MI 1條題 15分

when n = 1, LHS = 1 x 1 = 1
when n = 2, LHS = 2 x 1 + 1 x 2 = 4
when n = 3, LHS = 3 x 1 + 2 x 2 + 1 x 3 = 10
...
when n = k,
LHS = (k)(1) + (k - 1)(2) + (k - 2)(3) + ... + (3)(k - 2) + (2)(k - 1) + (1)(k)

when n = k + 1, LHS
= (k + 1)(1) + (k)(2) + (k - 1)(3) + ... + (3)(k - 1) + (2)(k) + (1)(k + 1)
= (k)(1) + (1)(1) + (k - 1)(2) + (1)(2) + (k - 2)(3) + (1)(3) + ...
+ (2)(k - 1) + (1)(k - 1) + (1)(k) + (1)(k) + (1)(k + 1)
= (k)(1) + (k - 1)(2) + (k - 2)(3) + ... + (3)(k - 2) + (2)(k - 1) + (1)(k)
+ (1)(1) + (1)(2) + (1)(3) + ... + (1)(k - 1) + (1)(k) + (1)(k + 1)
= (1/6)(k)(k + 1)(k + 2) + [1 + 2 + 3 + ... + (k + 1)]
= (1/6)(k)(k + 1)(k + 2) + (1/2)(k + 1)(k + 2)
= (1/6)(k + 1)(k + 2)(k + 3)
= RHS