(a) Use mathematical induction to show that for all n ≥ 1, (7 points) ( 1 2 ) + ( 1 2 )2 + ( 1 2 )3 + • • • + ( 1 2 )n =
Posted: Wed May 11, 2022 12:42 pm
(a) Use mathematical induction to show that for all n ≥ 1, (7
points) ( 1 2 ) + ( 1 2 )2 + ( 1 2 )3 + • • • + ( 1 2 )n = 1 - ( 1
2 )n, n ≥ 1
(b) Provide a direct proof of the following result. (8 points)
1* (1!) + 2 * (2!) + 3* (3!) + • • • + n* (n!) = (n + 1)! – 1.
(Hint. Write 1* (1!) = (2 - 1) * (1!) = 2! – 1! 2 * (2!) = (3 - 1)
* (2!) = 3! – 2! 3* (3!) = (4 - 1) * (3!) = 4! – 3! etc.)
points) ( 1 2 ) + ( 1 2 )2 + ( 1 2 )3 + • • • + ( 1 2 )n = 1 - ( 1
2 )n, n ≥ 1
(b) Provide a direct proof of the following result. (8 points)
1* (1!) + 2 * (2!) + 3* (3!) + • • • + n* (n!) = (n + 1)! – 1.
(Hint. Write 1* (1!) = (2 - 1) * (1!) = 2! – 1! 2 * (2!) = (3 - 1)
* (2!) = 3! – 2! 3* (3!) = (4 - 1) * (3!) = 4! – 3! etc.)