In Exercises 1-15 use mathematical induction to establish the formula for n ? 1. 1. 1. 2. + 2. 2. + 3. 2. + ··· + n. 2. = n(n + 1)(2n + 1).
?n is always divisible by 30. Use the fact that n. 5. ?n = (n?1)n(n+1)(n. 2. +1) to prove that it is divisible by 2 and 3 as well as 5.
1 · 3+2 · 4+3 · 5 + ··· + n(n + 2) = n(n + 1)(2n + 7). 6 is valid. Proof: (formal style; it is good to do a few proofs this way) We will use the Principle
12-Feb-2006 Example 3: for n a natural number prove that: 1) if n ? 2 then n3 ? n is always divisible by 3
2! + t4. 4! 3 t6. 6! + 17 t8. 8!
* Prove that. 12 + 22 + ··· + n2 = n(n + 1)(2n + 1). 6 for all positive integers n. Exercise 1.2. * Show that. 1 ?. 1. 2. +. 1. 3. ?
8.1.3 Some important observations. 1. The total number of terms in the binomial expansion of (a + b)n is n + 1 i.e. one more than the exponent n. 2.
If we had shown P(3) as our basis step then the inequality would only be proven for n ? 3. 2. For any positive integer n n. ? i=1. 1 i(i + 1). = 1. 1 · 2.
(a ? 1)(b ? 1)(c ? 1) = abc. ?=(a ? 1)2 ? 4a(a ? 1) = (1 ? a)(1 + 3a). ... (i) f(2n ? t(m)) ? 3(n?1)/2 if 2n + m is divisible by 3;.
principle of induction P(n) is true for all positive integers n > 1. herefore each term in the product (1+2+22)(1 +3 +3² +3³)(1 +.
So now for instance x2 ?2 is not divisible by x? ? 2 over Q since x? ? 2 doesn’t exist over Q Problem 9 Show that x2?2 is irreducible over Q (Hint: Suppose it were reducible What would the degrees of the factors have to be and what does that mean?) Problem 10 Show that x3 ?2 is irreducible over Q (Hint: This is similar
1+n2 = 0 and (ii) the sequence of terms 1+n2 are decreasing To see (i) notice that we can divide numerator and denominator by n2 to get lim n?? 1 n2 ·n 1 n2 (1+n 2) = lim n?? 1 n 1 n2 +1 = 0 To see (ii) let f(x) = x 1+x2 Then f0(x) = (1+x2)·1?x·2x (1+x2)2 = 1?x2 (1+x2)2 ? 0 for x ? 1 Therefore f is a decreasing
3 MATHEMATICAL INDUCTION 89 Which shows 5(n+ 1) + 5 (n+ 1)2 By the principle of mathematical induction it follows that 5n+ 5 n2 for all integers n 6 Discussion In Example 3 4 1 the predicate P(n) is 5n+5 n2 and the universe of discourse
Mathematical induction can be used to prove that a statement about n is true for all integers n ? 1. We have to complete three steps. In the basis step, verify the statement for n = 1. In the inductive hypothesis, assume that the statement holds when n = k for some integer k ? 1.
Therefore, f is a decreasing function in the relevant range, so the terms f(n) =n 1+n2are decreasing. 1 We know that the series converges, but we need to determine whether it converges absolutely or not. In other words, we must determine if X? n=1 (?1) nn 1+n2 = X? n=1 n 1+n2 . converges or not.
Definition 1Let p and q be polynomials in the complex numbers C. We say that p is divisble by q over C if there exists a polynomial r such that p(z) = q(z)r(z). Problem 1 By long division (which we learned how to do last week), answer the following: • Is x3?2x2+ x?2 divisible by x?2 over C?