# Use Mathematical Induction to Prove That the Statements are true for Every Positive Integer n. 1*3+2*4+3*5+â‹¯+n(n+2)=n(n+1)(2n+7)/6

### Solution:

First, let’s make sure we understand what we mean by “mathematical induction”: “Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one.” (Wikipedia) We have pasted below another more detailed explanation of how to create a proof using mathematical induction (see “Appendix” below).

Let’s now take the equation you provided: 1*3+2*4+3*5+…+n(n+2)=

We can show that this is true for n=1: 1*3 = = = 3

Now let us assume that the statement is true for n = k. If it is, then we will prove that it has to be true for n=k+1:

WTS è = +

* *

= +

=

=

=

=

= **QED**

In the above proof, WTS means “want to show” and QED means “quod erat demonstratum” (“which was to be demonstrated”).

## Appendix

Here’s another more detailed explanation of how to create a proof using mathematical induction:

**Related Topics**

- A Jeweler Needs to mix an Alloy with 16% Gold Content & an Alloy with a 28% Gold Content to Obtain 32oz. of a new Alloy with a 25% Gold Content. How Many oz. of Each of the Original Alloys Must be Used?
- If Utility Function is Expressed as U(x,y) = x^0.5 y^0.5 What is the Marginal Utility at Point (64,25) and (49,36)? Treat Y as a Constant. Would the Answer be MUx(x,y) = y. MUx(64,25) = 5 ? and MUx(49,36) = 6?
- N-3/8=6
- -2x < 3/7
- Solve the Triangle. Round Angle Measures to the Nearest Minute & Side Measures to the Nearest Tenth. a=11.4, b=13.7, c=12.2
- What Experiment Have 3 Types of Variables?
- What is the Volume of a Sphere the Radius of Six?
- Cos 2x=0.32
- Is 288/1155 in Lowest Terms
- What is the Largest Amount of Postage That Cannot be Made Using These Stamp Denominations: 17 Cents, 9 Cents & 5 Cents?

Posted in

**College, Homework Answers, Math Answers**.

This entry was posted on **Monday, September 6th, 2010** at **8:36 am** and is filed under **College, Homework Answers, Math Answers**. You can follow any responses to this entry through the **RSS 2.0** feed.
You can **leave a response**, or **trackback** from your own site.