Proof That The Sum The First N Odd Integers is a Square

The sum of the first n odd integers is a square. This is a theorem and can easily be proven if you have already learned proof by mathematical induction. Even though it sounds like a boring theorem, it is actually very interesting if represented visually. Consider the following diagrams and their numerical representations.

1+3=4

$latex 1 + 3 = 2^2$

 1+3+5=9

$latex 1 + 3 + 5 = 3^2$

Can you see the pattern? Let’s have some more examples.  Continue reading

The General Formula of Polygonal Numbers

When objects are arranged in certain ways, they  form a pattern. For example, square numbers can be obtained by forming objects in square arrangements. As shown below, the first four square numbers are 1, 4, 9, 16 and 25.

squarenumbers

Clearly, the number of dots in the 100th square number is $latex 100^2$. In general, the formula for finding the nth square number is $latex n^2$.

polygonal numbers

Now, how do we extend the formula of polygonal numbers? For example, what is the 6th pentagonal number (shown in the figure above) and the 10th  heptagonal number? Try to see if you can see a pattern from the second figure. If you can derive the formula for the triangular numbers, it will be easier for you to solve the problem. You can also answer the following questions.  Continue reading

A Geometric Proof of an Infinite Series

In the previous post, we have seen how easy it is to prove a problem in Geometry using Algebra. The problem could also be proven geometrically, but the proof is longer. In this post, we will learn how to use Geometry to prove a problem on infinite series. That is, we have to show geometrically, the sum of

$latex \displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots$.

Note that the symbol $latex \cdots$ means that the number of terms is infinite; that is, the addition continues without end.  In this post, we show that the sum of this infinite series is 1.

Theorem

$latex \displaystyle\frac{1}{2} + \displaystyle\frac{1}{4} + \displaystyle\frac{1}{8} + \displaystyle\frac{1}{16} + \displaystyle\frac{1}{32} + \displaystyle\cdots = 1$

Continue reading