#1**+1 **

1.

n^3 = ( n - 1)^3 +( n - 2)^3 + ( n - 3)^3

Simplify

n^3 = 3n^3 -18n^2 + 42n - 36 rearrange as

2n^3 - 18n^2 + 42n - 36 = 0 divide throught by 2

n^3 - 9n^2 + 21n - 18 = 0 [ split up the 9n^2 term as 6n^2 + 3n^2 ]

n^3 - 6n^2 + 3n^2 +21n - 18 = 0 this factors as

n^2 ( n - 6) + 3(n^2 + 7n - 6) = 0

n^2(n - 6) + 3 (n - 6) ( n + 1) = 0

(n - 6 ) [ n^2 + 3(n + 1) ) = 0

(n - 6) [ n^2 + 3n + 3] = 0

The second factor has no real solutions when set to 0 [ the discriminant is negative ]

So

n - 6 = 0

n = 6

CPhill May 24, 2019