Induction Problems: Solving (n^2-n)/2 & sqrt(n) < 1/sqrt(1)+...+1/sqrt(n)

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In summary, the conversation discussed two problems. The first problem involved proving that the number of line segments joining n points in a plane with no 3 points colinear is (n^2-n)/2. The second problem involved proving that sqrt(n) is strictly less than 1/sqrt(1) +1/sqrt(2)+...+1/sqrt(n) for n\geq 2. The attempt at a solution for the first problem involved considering the addition of a new point and determining the number of new lines that would need to be drawn. The attempt at a solution for the second problem involved manipulating the equation and substituting in the induction assumption, but the speaker was unable to make progress with this approach.
  • #1
guroten
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Homework Statement


show that for n points in a plane, with no 3 points colinear, the number of line segments joining all pairs of points is (n^2-n)/2


Problem 2
Show that sqrt(n) is strictly less than 1/sqrt(1) +1/sqrt(2)+...+1/sqrt(n) for n[tex]\geq[/tex] 2


The Attempt at a Solution


For problem 1, I have no idea how to start. For problem 2, I tried manipulating the equation and substituting in the induction assumption, but I couldn't get anywhere with it.
 
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  • #2
Suppose you already have n points all joined up and you add a point somewhere else not joined to any other point. How many lines do you have to draw to connect this one point to every point in the existing diagram?
 
  • #3
I figured out the first problem, but I'm still having trouble with the second. Any suggestions?
 
  • #4
You haven't shown us what you did, so we don't know what your problem is. What do you need to prove to make the induction work?
 

Related to Induction Problems: Solving (n^2-n)/2 & sqrt(n) < 1/sqrt(1)+...+1/sqrt(n)

1. What is the purpose of solving induction problems?

The purpose of solving induction problems is to prove a mathematical statement or formula for all possible values using a specific method called mathematical induction.

2. What does (n^2-n)/2 represent in the given induction problem?

In the given induction problem, (n^2-n)/2 represents the sum of the first n consecutive even numbers.

3. How do you prove (n^2-n)/2 using mathematical induction?

To prove (n^2-n)/2 using mathematical induction, we first show that the statement is true for the initial value of n (usually n=1). Then, we assume that the statement is true for some value k, and use this assumption to prove that the statement is also true for k+1. Finally, we conclude that the statement is true for all possible values of n.

4. What is the significance of sqrt(n) < 1/sqrt(1)+...+1/sqrt(n) in this induction problem?

The inequality sqrt(n) < 1/sqrt(1)+...+1/sqrt(n) is a crucial part of this induction problem as it helps us to establish the base case and to prove the induction step. It also demonstrates the concept of convergence in mathematical induction.

5. Can induction problems be solved using other methods besides mathematical induction?

Yes, induction problems can also be solved using direct proof, proof by contradiction, and proof by contrapositive. However, mathematical induction is the most commonly used method for solving these types of problems.

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