
PROVE BY INDUCTION
Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by...
11. Let an >0 and assume that bn = n+1 + B. What can we say about the convergence of an? an
Construct a PDA that accepts {a"ba" bn | n0Am >0};
let a,b > 0 . Prove that
DI < Val
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 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
Suppose that an >0 and bn >0 for all n2N (N an integer). If lim = , what can you conclude about the convergence of an? A. a, diverges if by diverges, and an converges if bn converges. an diverges if by diverges. c. a, converges if be converges. OD. The convergence of an cannot be determined.
Prove by mathematical induction (discrete mathematics)
n? - 2*n-1 > 0 n> 3
Prove that is an integer for all n > 0.
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.