A Geometric Model: The ancient Greeks often discovered mathematical relationships by using geometric drawings. Study the accompanying drawing and use inductive reasoning to determine what needs to be put in place of the question mark the equation a true statement. How does the sum of the first odd integers related to the geometric model? Is it possible to find the sum of the first even integers using geometric model? Why or why not?

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A Geometric Model: The ancient Greeks often discovered mathematical relationships by using geometric drawings. Study the accompanying drawing and use inductive reasoning to determine what needs to be put in place of the question mark the equation a true statement.  How does the sum of the first odd integers related to the geometric model? Is it possible to find the sum of the first even integers using geometric model? Why or why not?

crocolylec · Accepted Answer

Step 1  To check: Whether the statement is true or not by inductive reasoning  Given information:  The statement is 1+3+5+7+⋯(2n−1)=?  Explanation:  It can be seen from the geometric figure, that it is started with 1 circle.  Then after that if we add a layer of 2+1 layers, then it got arranged to 2 by 2 square.  Now in the square, there is 4 circles.  Now add a layer of 3+2 circle in the resultant square.  We get a square 3 by 3 square that holds 9 circles.  Similarly, in same fashion,  Inductive reasoning will take us to the conclusion the last addition of a layer that would be (n by n) square.  The question mark can be replaced by n2  Therefore, 1+3+5+7+⋯(2n−1)=n2  Step 2  To determine: To sum of first even integers by using geometric model.  Explanation:  It can be seen that there is addition of odd integers in the geometric model.  In even integers, the scenario will be,  1) 1(1+1)=1×2=2  2) 2(2+1)=2×3=6  3) 3(3+1)=3×4=12  and so on.  We get the last term n(n+1)  That is not possible to calculate as per the by geometric model.

A Geometric Model: The ancient Greeks often discovered mathematical re

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