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Solution :-
Prove that is an integer for all n > 0.
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,
Use the Principle of Mathematical Induction to prove that (2i+3) = n(n + 4) for all n > 1.
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
12. Prove that if n >m then the number of m-cycles in Sis given by nn-1)(n-2)... (n-m+1)
Use induction and Pascal's identity to prove that (7) = 2" where n > 0.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
2. Prove that if n > 1, then 1(1!) + 2(2!) + ... + n(n!) = (n + 1)! - 1.
Problem 3. Find the exact solutions to the following recurrences and prove your solutions using induction 1, T(1) = 5 and T(n) T(n-1) + 7 for all n > 1. 2. T (1)-3 and T(n)-2T(n-1).