Blog 1- Searching for Patterns in Mathematics

Mathematics is about finding and collecting the existing patterns from what we have learned or in our daily lives, then combining all materials to summarize as a definition, a theorem or even a formula.

We can start with a simple expansion below.

(a + b) (a – b) = (a + b)a – (a + b)b

                     = aa + ba – ab – bb

     = a­² – b²

Now, we want to substitute 1 and x for a and b, respectively, we have

(1 + x) (1 – x) = 1 – x²

Now let’s replace (1 + x) in (1 + x)(1 – x) with (1 + x + x²), we obtain that

(1 + x + x²) (1 – x) = (1 + x + x²)1 – (1 + x + x²)x

                          = 1 + x + x² – (x + x² + x³)

                               = 1 + x + x² – x – x² – x      

= 1 – x³

We are catching some patterns here. Only the left and right ends have left, those middle terms got canceled out. After testing examples with the same kind of expansion, we can generalize them for all integer n

(1) (1 – x) = 1 – x

(1 + x) (1 – x) = 1 – x²

(1 + x + x²) (1 – x) = 1 – x³

(1 + x + x² + x³) (1 – x) = 1 – x⁴

(1 + x + x² + x³ + x⁴) (1 – x) = 1 – x⁵

. . .

(1 + x + x² + x³ + x⁴ + . . . + xⁿ) (1 – x) = 1 – x^{n + 1} (*)

We will assume 1 – x does not equal 0 to avoid the division errors, divide both sides of equation (*) by (1 – x), and conclude that

We have obtained a formula for the sum of the first n terms of the geometric series {1, x, x², x³, …, xⁿ, …}. And we have known that the series go on forever. Therefore, when |x| < 1, then x^{n + 1}  approaches 0 as n approaches infinity, we deduce 

That is how we generated the formula for a geometric series. 

Mathematics never disappoints us, it brought us from one surprise to another surprise, and many more to go. It's always like a long adventure to go on and discover something new. In math, most of everything has a pattern, you just have to catch it, do some algebras, make conditions, and figure out how far it can get you to.