(4) (8 marks) Prove by induction that if q is rational and n e N then q2+1 is also rational
2. [14 marks] Rational Numbers The rational numbers, usually denoted Q are the set {n E R 3p, q ZAq&0An= Note that we've relaxed the requirement from class that gcd(p, q) = 1. (a) Prove that the sum of two rational numbers is also a rational number (b) Prove that the product of two rational numbers is also a rational number (c) Suppose f R R and f(x)= x2 +x + 1. Show that Vx e R xe Qf(x) Q...
Prove using mathematical induction: (4) Prove that for all n E N, 3(7" – 4”).
Q (8 points) Use mathematical induction to prove the formula 1 X – 1 1 X x(x – 1) 22 2n for all n = 1, 2, 3, ..., and x + 0,1.
Induction proofs. a. Prove by induction: n sum i^3 = [n^2][(n+1)^2]/4 i=1 Note: sum is intended to be the summation symbol, and ^ means what follows is an exponent b. Prove by induction: n^2 - n is even for any n >= 1 10 points 6) Given: T(1) = 2 T(N) = T(N-1) + 3, N>1 What would the value of T(10) be? 7) For the problem above, is there a formula I could use that could directly calculate T(N)?...
4. Prove by induction that for r 1 1- n+1
8. Prove that if two rational sequences (a)1 and (n)1 are equivalent, then (a) (an) is Cauchy if and only if (bn) is Cauchy. (b) (an) is bounded if and only if (%) is bounded. 8. Prove that if two rational sequences (a)1 and (n)1 are equivalent, then (a) (an) is Cauchy if and only if (bn) is Cauchy. (b) (an) is bounded if and only if (%) is bounded.
Please Prove. Prove 2 n > n2 by induction using a basis > 4: Basis: n 5 32> 25 Assume: Prove:
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
11. Prove Bonferroni's ineqyuality r (n A) PLA)-(1) by induction n 1) by induction.
7n Use Mathematical Induction to prove that Σ 2-2n+1-2, for all n e N