
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,
(5) Use induction to show that Ig(n) <n for all n > 1.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
5. Prove that U(2") (n > 3) is not cyclic.
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
(1) Prove that for Res > 3 (s)S(s - 2)- n-1 where σ2(n)-2.1n d2 is the sum of the squares of the positive divisors of n.
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
Prove each problem, prove by induction
3) Statementn-1 5 25(2m-1) forn2 1 4 Statement Suppose: bo1 . b,-2b-1 + 1 for t 1 en fort >
Prove that is an integer for all n > 0.