Question

DISCRETE MATHEMATIC

For question 1, Use mathematical induction to prove the statements are correct for n ∈ Z+(set of positive integers).

1. Prove that for n ≥ 1

1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2

For question 2, Use a direct proof, proof by contraposition or proof by contradiction.

2. Let m, n ≥ 0 be integers. Prove that

if m + n ≥ 59 then (m ≥ 30 or n ≥ 30).

Answer 1

1) n>=1, 1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2

Base case: n = 1, [n(7n - 5)]/2 = 1*2/2=1. The statement 1=1 is true.

Assume for n the statement is true, so 1 + 8 + 15 + ... + (7n - 6) = [n(7n - 5)]/2

For n+1 the series will 1 + 8 + 15 + ... + (7n - 6) + (7*(n+1)-6) => [n(7n - 5)]/2 + 7n+1 => (7n^2 - 5*n + 14*n + 2)/2 = (7*n^2+9*n+2)/2

[(n+1)(7*(n+1) - 5)]/2 => (7*(n+1)^2 - 5*n -5)/2 => (7(n^2+2*n+1)-5*n-5)/2 => (7(n^2)+14*n-5*n+7-5)/2 => (7*n^2+9*n+2)/2

We can see that 1 + 8 + 15 + ... + (7n - 6) + (7*(n+1)-6) =>[(n+1)(7*(n+1) - 5)]/2

Hence proved

2) m+n>=59 => m >= 59 - n and n >= 59 - m

lets assume m<=29 => -m >= -29 => 59 - m >= 30

Combining n >= 59 - m and 59 - m >= 30 we get n>=30

Lets assume n <=29 => -n >= -29 => 59 - n >= 30

Combining m >= 59 - n and 59 - n >= 30 we get m >= 30

Hence proved that m>=30 or n>=30.