10. Math 213 Worksheet: Induction Proofs A.J. Induction proofs, type I: Sum/product formulas: The most common, and the easiest, application of induction is to prove formulas for sums or products of n terms. Proof by (Weak) Induction When we count with natural or counting numbers (frequently denoted N {\displaystyle \mathbb {N} } ), we begin with one, then keep adding one unit at a time to get the next natural number. Note - a convex polygon For our base case, we need to show P(0) is true, meaning the sum of the first zero powers of two is 20 – 1. Proof: By induction.Let P(n) be “the sum of the first n powers of two is 2n – 1.” We will show P(n) is true for all n ∈ ℕ. Sometimes a proof by induction will obscure such an understanding. Mathematical Induction is a proof technique that allows us to test a theorem for all natural numbers. Theorem: The sum of the first n powers of two is 2n – 1. Give a formal inductive proof that the sum of the interior angles of a convex polygon with n sides is (n−2)π. Now we have an eclectic collection of miscellaneous things which can be proved by induction. Multiplying both sides with a−1 gives We’ll apply the technique to the Binomial Theorem show how it works. We watch way too much television and are content to accept things as true without question. Section 1: Induction Example 3 (Intuition behind the sum of ﬁrst n integers) Whenever you prove something by induction you should try to gain an intuitive understanding of why the result is true. The simplest application of proof by induction is to prove that a statement P(n) ... and using the induction hypothesis, the sum in the left hand side can be expressed using the formula. Hildebrand Practice problems: Induction proofs 1. You may assume that the result is true for a triangle. Uses worked examples to demonstrate the technique of doing an induction proof. Prove that T n < 2n for all n 2Z +. Thus, we need to prove an+1 + an+1 −1 a−1 = an+2 −1 a−1. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Hence, a single base case was su cient. All of these proofs follow the same pattern. In the following array, you will ﬁnd one 1, two 2’s, three 3’s, etc. 7.4 - Mathematical Induction The need for proof. The sum, S n, of the first n terms of an arithmetic series is given by: S n = ( n /2)( a 1 + a n ) On an intuitive level, the formula for the sum of a finite arithmetic series says that the sum of the entire series is the average of the first and last values, times the number of values being added. 37. directly to the n = k case, in the same way as in the induction proofs for summation formulas like P n i=1 i = n(n+ 1)=2. Most people today are lazy. Since the sum of the first zero powers of two is 0 = 20 – 1, we see The sum of a constant times a function is the constant times the sum of a function. Let the \Tribonacci sequence" be de ned by T 1 = T 2 = T 3 = 1 and T n = T n 1 + T n 2 + T n 3 for n 4. The sum of a sum is the sum of the sums ∑(x+y) = ∑x + ∑y.

2020 proof by induction summation