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Question 19 Prove the following statement: 1.1!+2 2!+...+d. d! = (d + 1)! – 1 when...
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
Prove by mathematical induction (discrete mathematics) n? - 2*n-1 > 0 n> 3
Prove that is an integer for all n > 0.
Problem 1 [8pt] Prove that the following two Hoare triples are valid. (Hint: in predicate logic Pi equivalent to -P V P2) is a) (4pt) y:= x * 2; y:= y + 3; lu > o) b) (4pt) if (y>2) x := y-1; elsex5-y: fr > 1)
Find the rectangular coordinates for the point whose polar coordinates are given. 8 TT 6 (x, y) = ) =( Convert the rectangular coordinates to polar coordinates with r> 0 and 0 se<2n. (-2, 2) (r, 0) Convert the rectangular coordinates to polar coordinates with r> 0 and O So<211. (V18, V18) (r, ) = Find the rectangular coordinates for the point whose polar coordinates are given. (417, - ) (x, y) =
Fact: If d > 2 is an integer, then there exists a prime q such that q divides d. (1) Let e and f be positive integers. Prove that if ged(e, f) = 1, then god(e?,f) = 1. (2) Let m be a positive integer. Prove that if m is rational, then m is an integer.
Exercise 1.25. This exercise relates to (1.13). Suppose that x > 1. For each ne N let y= Vå be defined by yn = x. This implies that vx < mă if n > m. To prove that Veso 3NEN Vnən : 0< V2-1<e it therefore suffices to prove that Vesonen: 0< Vx-1<e. Prove this latter statement.
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
3. Use the mean value theorem to prove the following inequality. (1 +x)" >1 for z >0 andnEN
| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.