5. For +define the sum by the formula = 1 2! + 2 3! + 3 4! + + ( 1) ! + ( + 1)!

a. Compute first five terms.

b. Depending on the results you have found in part (a), conjecture a formula for the sum of the terms in .

c. Verify your conjecture in part (b) for all + by mathematical induction.

6. Solve the recurrence = 31 with initial condition 0 = 2 using generating functions

5. Forn e Z define the sum s, by the formula (n-1) n 1 + 2! 2 3 + 3: 4! (n+1)! a Compute first five terms. b. Depending on the results you have found in part (a), conjecture a formula for the sum of the terms in se c. Verify your conjecture in part (b) for all ne zby mathematical induction. 6. Salve the recurrence c = 30,-- with initial condition , = 2 using generating functions