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

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Homework Help Overview

The discussion revolves around two problems involving combinatorial geometry and inequalities in sequences. The first problem concerns the number of line segments that can be formed by connecting pairs of points in a plane, while the second problem involves proving an inequality related to the sum of the reciprocals of square roots.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the combinatorial reasoning behind counting line segments in the first problem and question how to approach the second problem involving inequalities. One participant suggests considering the addition of a new point to an existing configuration to analyze the first problem further.

Discussion Status

The discussion is ongoing, with some participants expressing uncertainty about their approaches. One participant has made progress on the first problem but is seeking further assistance on the second. There is a call for clarification on the steps taken to apply induction in the second problem.

Contextual Notes

Participants are working under the constraints of homework rules, which may limit the sharing of complete solutions. There is an emphasis on understanding the reasoning behind the problems rather than simply arriving at answers.

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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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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?
 
I figured out the first problem, but I'm still having trouble with the second. Any suggestions?
 
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?
 

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