Sum of an Infinite Series using Equilateral Triangle

Mathematics makes a nice distinction between the usually synonymous terms “elementary” and “simple”, with “elementary” taken to mean that not very much mathematical knowledge is needed to read the work and “simple” to mean that not very much mathematical ability is needed to understand it.

Julian Havel

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Consider a finite series, say of n terms. We can find the sum by adding the terms or using known formula. For example, to find the sum of first 100 natural numbers, we can go on adding them to find the sum or we can apply the known algebraic formula and in any case the sum will be 5050.

The same is not possible in case of an infinite series, because we cannot go on adding the terms.  Known algebraic formulas are there to find the sum. Sometimes, geometric reasoning helps us to find the sum. Here we will consider the following infinite series and will find it’s sum.

Of course, we can solve it using the concept of infinite geometric progression. Here a = 1/4 and r = 1/4. Therefore, required sum of the infinite series is

Apart from this algebraic formula, there is an excellent way to find the sum of this infinite series. The idea is to use the concepts of geometry (in fact, using equilateral triangles).

Consider the above diagram, where ABC is an equilateral triangle. Assume that X, Y and Z are the mid-points of the sides BC, AB and CA. Also, P, Q, R are the mid-points of the sides AY, ZA and YZ. Clearly, XYZ is also an equilateral triangle, as it is formed by joining the mid-points. In a similar way, PQR is also an equilateral triangle. Let the area of the triangle ABC be M. Note the following points:

(i) All of the triangles AYZ, BXY, CXZ are congruent and hence they all are equal in area. Therefore, area of the largest blue triangle XYZ is equal to 1/4 th the area of the triangle ABC, or (1/4)M.

(ii) Similarly, all of the triangles YPR, ZQR, APQ are congruent and hence they all are equal in area. Therefore, area of the next largest blue triangle PQR is equal to 1/4 th the area of the triangle AYZ. But the area of the triangle AYZ is equal to 1/4 th the area of the triangle ABC. Hence, area of the triangle PQR is 1/4 of 1/4 , or 1/16 of the area of the triangle ABC or (1/16)M.

(iii) Continuing in this way, we find the area of the next largest equilateral triangle is 1/64 of the area of the triangle ABC or (1/64)M; next one is 1/256 of the area of the triangle ABC or (1/256)M and so on.

So, the sum of all the equilateral triangles (blue colour) is given by

Now, for every blue triangle, there is a congruent green triangle left to it and a congruent red triangle right to it. When we add the area of all such blue, green and red triangles, we will get the area of the whole triangle ABC. But, area of the triangle ABC is M (a non-zero quantity). Therefore, area of the blue triangles is M/3 (also for green and red triangles). Hence

Since M is a non-zero quantity, it follows that

as expected!

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