(2) Using the identity: n! k!(n - k)! for n > 2, prove that the following identity is even: 1 n
(3) Using the identity: (*) – 16–191 n! k!(n-k! k for n > 2, prove the following identity: (n-2 + (5+1) 1
1) Give a combinatorial proof of the following identity (0 <k<n): n2 k ---- = n.29-1 ke=0
Use induction and Pascal's identity to prove that (7) = 2" where n > 0.
2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n +
n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2
for all n ≥ 1. (b) Use (a) and the -definition of limit to show
that limn→∞ xn = 0.
Exercise 2. Consider the sequence (In)n> defined by cos(k)...
2. Use induction to prove that the following identity holds for al k 2 (n 1)2"+12 Be sure to clearly state your induction hypothesis, and state whether you're using weak induction or strong induction
3. Give a combinatorial proof of the following identity. ("t?) = () + (-1) where n and k are positive integers with n > k. 5. Give a combinatorial proof of the following identity (known as the Hockey Stick Identity). (%) + (**") + (**?) + ... + ( )= (#1) where n and k are positive integers with n > k.
19. Recursively define co = 5, Ck = (Ck-1)2 for k > 1. Prove using induction that for n 20, Cn = 52". Note that in the explicit formula for C, the exponent of 5 is 2".
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
2. Use Method of mathematical induction to prove identity : for all natural n > 2 1.1+(1.1)? + ... + (1.1)n-1 = - 11n-1 1.1 - (1.1)" - 0.1 inf of the set below