Problems on proof techniques

Here are 25 proof problems. Each 5 correctly solved problems will improve your Test 02 grade by one level. E.G. if you solve 5 problems and if your grade was B-, the improved grade will be B. Solve 10problems and you will receive 2 letter grade levels increase of your Test 02 grade. Solve all problems and you will receive 5 levels increase.

Choose and apply a method of proof or disproof (direct proof, proof by contraposition, proof by contradiction, disprove by counterexample, proof by mathematical induction) to prove or disprove the following statements:

  1. For all integersn, n2is even if and only if nis even. Note: if and only if means: a) ifn2is even thennis even, b) ifnis even thenn2is even
  2. For all integersk, k3is odd if and only if kis odd
  3. For all prime numbers, the sum of any two prime numbers each larger than 2 is not a prime number.
  4. For allxrational and for allyirrational,x + yis an irrational number. Note: xis a rational number if and only if there exist two integersaand b, b  0, such thatx = a / b.
  5. 2 + 4 + 6 + …. + 2n = n(n+1) , for all n ≥1
  6. 12 + 32 + 52 + …. + (2n-1) 2 = n(2n+1)(2n-1)/3, n ≥1
  7. 1 + a + a2 + a3 + …. + a(n-1) = (an – 1)/(a-1), n ≥1, a  1
  8. For all integersn, ifnis even then(n-1)(n+1)is odd
  9. For all integersn, if(n+1)(n-1)is odd thennis even
  10. Consider the sequence a1, a2, …, an …. defined recursively:

a1 = a, an+1 = an + d, n ≥1, d  0

Prove that an = a1 + (n-1)d

  1. The square of any integer can be written in one of the following forms:4k or 4k+1
  2. Letnbe an odd integer. Then n3 + 2n2is also odd.
  3. The sum of any two rational numbers is rational
  4. The product of any two irrational numbers is irrational
  5. The sum of any two irrational numbers is irrational
  6. 2 + 6 + 18 + …. + 2*3(n-1) = 3n – 1, n ≥1
  7. 1*2 + 2*3 + 3*4 + …. + n(n+1) = n(n+1)(n+2)/3
  8. xn – 1 is divisible by x-1 for n ≥1, x  1
  9. 32n + 7 is divisible by 8, n ≥0
  10. (1 – ½)(1 – 1/3)…..(1 – 1/(n+1)) = 1/(n+1) for every positive integern
  11. n2 < 2nfor every integer n ≥ 5
  12. n2 > 2n + 1 for n ≥ 3
  13. 7n – 2nis divisible by 5 for all n ≥1
  14. n! > n3for all n ≥ 6
  15. (a/b)(n+1) < (a/b)nfor n ≥1 and 0 < a < b