Prove that ( n2 + 1) / (n + 1) diverges to +∞

consider the statement (formula) (3x)A(x) → A(z) where z is a new variable not free (not an "input variable") in A(x). Find now a specific example of A(x) over the set N and choose a specific value of z N so that (1) becomes false (meaning we cannot prove it, since proofs start from true axioms and preserve truth at every step). (1)

part ii and iii, thank you

Use induction to prove: for any integer n ≥ 1,(6-4)= 3n²-n. Base case n = Inductive step k+1 j=1 Assume that for any k > " = = j=1 (6j-4)=(6j- 4)+ j=1 =( (6j-4)= = 3 k² + k²+ k+ k+ j=1 j=1 , 3n²-n= Σ(6j-4)= )-(k+ 1) -(k+ 1) we will prove that By inductive hypothesis k+1 j=1 (6j-4)=

Prove

Find the disjunctive normal forms of the following Boolean expressions (showing all the working) (a) (x + z) ' (y+x'z) (b) xz + (x'+y+z)' (c) (x + y) ⊕ (xz)

Expand In t(x²+1) (x+2) (x-s)² X-5

Find the general solution of the recursive relation an an = a10 (3)" + α20 (-3)" an = (a10+ a11 n) (-3)" None of these. an = α10 (6)" + α20 (-9)" an = (α10+ α11 n) (3)" = 6an-19an-2, n ≥ 2.

Use generating functions to solve an 2an 1+1 for n ≥ 1 where ao = 0. [At most 1 point if you don't use generating func- tions.] =

Prove that f(n) = 1000n5 + 20000n2 + 32 is O(n6 ); Is this a tight upper bound? Why or why not?