Proof：練習問題

To prove by mathematical induction that a statement is true for all integers \(n \ge 1\), which of the following is the required first step (the base case)?

To prove by contradiction that "There are no positive integers \(x\) and \(y\) such that \(x^2 - y^2 = 10\)", which initial assumption should be made?

A student is proving that \(\sum_{r=1}^n r(r+1) = \frac{n(n+1)(n+2)}{3}\) by mathematical induction. During the inductive step, the student adds the \((k+1)\)-th term to the sum of the first \(k\) terms. Which of the following shows the correct factorization of the resulting expression \(\frac{k(k+1)(k+2)}{3} + (k+1)(k+2)\)?

Consider the statement: "For all prime numbers \(p\), the value of \(p + 2\) is also a prime number."
Which of the following values of \(p\) provides a counter-example to this statement?

Consider the statement: "If \(n\) is a positive integer, then \(n^2 - n + 41\) is a prime number."
Which of the following values of \(n\) provides a counter-example to this statement?

Show, by means of a counter-example, that the statement "\(n^2 + n + 1\) is a prime number for all positive integers \(n\)" is false.

Prove by contradiction that if \(n\) is a positive integer, then \(n^2 + 2\) is not divisible by 4.

Given that \((x-1)\) is a factor of \(f(x) = 2x^3 + px^2 - 10x + q\), and that when \(f(x)\) is divided by \((x+2)\) the remainder is 25, find the values of the constants \(p\) and \(q\).

Use the method of mathematical induction to prove that for all positive integers \(n\),
\(\sum_{r=1}^{n} (3r - 2) = \frac{n(3n - 1)}{2}\).