Geometric Series

To find the sum S, of n terms of increasing powers:

S = 1 + r + r² + . . . + r^n-1

eliminate terms by evaluating the difference:

rS - S = (r + r² + r³ + . . . + rⁿ) - (1 + r + r² + . . . + r^n-1)

For an alternative solution, note that S is similar to the representation in base r of the number: 111...1  of n terms, each digit having the place value of a power of r.

If each digit of a number in base r had the value r-1, then adding 1 to this number would cause a carry to flow from the lowest digit to the highest, yielding a number with a 1 digit in the next higher place position followed by all 0 digits.  In a similar way:

(r-1)S = (r-1) + (r-1)r + (r-1)r2 + . . . + (r-1)n-1
  +        +       +        +                 +
  1    =   1  -->  r   -->  r2  --> . . . --> rn-1 -->  rn