Thanks for this help

Example 7. Use a generating function to verify the formula Sn = 1 + 4 + 9 +... + n2_ n(n + 1)(2n+ 1) Proof. We just saw that 6 f (2 ) = (1 -x)3 generates 0,1,4,9,16,..., hence g(x) x 2 + x (1 - 2) 4 generates Sn. In other words, Sn is the coefficient of x2 in g, which is the coefficient of x"-2 plus the coefficient of Non-1 in (1 - x)-4. So Sn = ( n2 - 1 ) + (n - 2) = n + 1) (n - 2) + (n + 2 n - 1 = (n + 1) + (n + 2 3 3 (n + 1)n(n - 1) + (n + 2)(n + 1)n 6 = (n + 1)n[(n -1) + (n+2)] 6 = n(n + 1) (2n + 1) 6 0 Exercise 8. Use the above method to verify that 1 + 23 + 33 + 43 + .n3 _ n(n+1)2 2