Mathematical Induction Sample Problems

For any integer n 1 let pn be the statement that 1 4 7 3n 2 n 3n 1 2.

Mathematical induction sample problems. Department of mathematics uwa academy for young mathematicians induction. Here we are going to see some mathematical induction problems with solutions. 3 prove that the sum of the first n non zero even numbers is n2 n. 4 make up your own induction problems in most introductory algebra books there are a whole bunch of problems that look like problem 1 in the next section.

Solution to problem 1. Prove that for any natural number n 2 1 2 2 1 3 1 n 1. Problems with solutions greg gamble 1. If you think you have the hang of it here are two other mathematical induction problems to try.

1 the sum of the first n positive integers is equal to n n 1 2 we are not going to give you every step but here are some head starts. Mathematical induction is a method or technique of proving mathematical results or theorems. Write the statement to be proved as p n where n is the variable in the statement and p is the statement itself. Mathematical induction tom davis 1 knocking down dominoes the natural numbers n is the set of all non negative integers.

Left side 1 right side 1 1 1 2 1 both sides of the statement are equal hence p 1 is true. They add up a bunch of similar polynomial terms on one side and. Hence 1 1 2 1 2 3 1 n 1 n 1 1 1 2 1 2 1 3 1 n 1 1 n 1 1 n n 1 n. Observe that for k 0 1 k 1 k 1 k 1 k k k 1 1 k k 1.

Example if we are to prove that 1 2 3 4. 1 1 n 2 n 1 2n. In these problems f n is a fibonacci number. I would not ask you to do a problem this hard in a test or exam.

We first show that p 1 is true. Let the statement p n be 1 2 3. The statement p1 says that 1 1 3 1 2. 12 22 32 n2 1 6 n n 1 2n 1 3.

1 1 2 2 1 1 3 2 1 1 4 2. Thus the formula is true for all n by the principle of induction. Problem 1 use mathematical induction to prove that 1 2 3. N n n 1 2 we say let p n be 1 2 3 4.

Define mathematical induction. 1 2 3 n 1 2 n n 1 2. First prove 1 1 2 1 2 3 1 n 1 n n 1 n. N n n 1 2 step 1.

A few are quite difficult. Here are a collection of statements which can be proved by induction. The solution in mathematical induction consists of the following steps. Mathematical induction worksheet with answers.

Fix k 1 and suppose that pk holds that is 1 4 7 3k 2 k 3k 1 2. Fibonacci fun there are literally dozens hundreds of formulas involving fibonacci numbers and some of them provide good practice in induction. Prove using mathematical induction that for all n 1 1 4 7 3n 2 n 3n 1 2. N n n 1 2 for all positive integers n.

Induction examples question 1. The difficult ones are marked with an asterisk. F n 1 f n 2 if n 2 and f 0. The process of induction involves the following steps.

By the principle of. Principle of mathematical induction examples.