Sum of Natural number


Take any 2 circles among the given blue circles.
Draw 2 lines inclined at $60 ^{\circ}$ to each other .
For every such pair of points we get a corresponding yellow point.
If the total number of blue circles is $n$,the number of ways to select 2 circles = $\binom{n}{2}$.
Count the total number of yellow circles.
Now I guess it is easy to see that:
$$1+2+\cdots + (n-1) = \binom{n}{2}$$

Comments

Post a Comment

Popular posts from this blog

Viviani's Theorem

Image
Viviani's Theorem states that for any point $\mathcal{P}$ inside an equilateral triangle $\mathcal{ABC}$ the sum of the lengths of perpendiculars drawn from P on the sides is equal to the length of the altitude.    Draw equilateral triangles from that point onto the sides of the triangle such that these 3 perpendiculars are the altitudes of the equialteral triangles.Now rotate the triangles as shown.Now it can be easily seen that the sum of the 3 perpendiculars=perpendicular of the  $\Delta ABC$  
Keep reading

Sum of odd numbers

Image
Proof of $$1+3+5+\cdots = n^{2}$$ Count the number of balls in the L shaped regions.Notice that they are odd numbers. Furthermore,count the number of balls considering the length or the breadth(they will turn out to be the same). Sum of all balls = (number of balls on the length)$^{2}$.
Keep reading