One way to prove the above equality is to use summation of geometric progression which goes as follows,
My intention is to get (xⁿ – 1) from scratch with a unique way, by changing base of natural numbers and exploiting their ease of usage.
The number system we generally use is base 10, ie., using a permutation of 10 different digits from 0 to 9 and their known place values. But base 2 numbers use 2 digits only ie., 0 and 1. Further we can generate any base number system. I will represent base 2 number as , base 10 numbers as and generally base (n + 1) numbers as
In base 2 number system following observation can be done,
(I feel the below conclusion is obvious for some readers)
Lets consider the case of n digit base 2 number,
Hence, equation for is shown for x = 2. Lets consider n digit base 3 number,
Hence, equation for is shown for x = 3. Lets consider n digit base x number,