2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
5. If a, b E R, prove that abl < (a + b^).
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Let U ? Rmxn. Prove that if UTI-In, then n < m.
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.
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.
k=42, m=18 n=4 11. Let F:R → R be a function such that (t+m)(n+1) (n+ m F(t) = for t <-m, f or-m <t<n. for n<t<k, for t > k. nA - 1 Find A and B knowing that F is the cumulative distribution function of a random variable X such that P(X = k) = . Please provide only the value of parameter B in the space specified below. ANSWER: B= Solution:
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
PROVE: 4. If f : R → R is a strictly increasing function, f(0) = 0, a > 0 and b > 0, then