In an earlier post on manipulating fractions, I mentioned the useful addendo property: if a/b = c/d, then
This property is in fact used in Euclid’s Elements (Book 9, Proposition 35) written some 2300 years ago to sum a geometric series!
The usual way one is taught to find the sum
is to multiply both sides by r to give
and then subtract the first equation from the second:
Here is an alternative approach based on Euclid’s work. We begin with the ratios
Subtracting 1 from each of these ratios (equal to r) gives
By addendo each of these ratios (equal to r-1) is also equal to
The numerator here simplifies greatly and we are left with
Geometric sequences have a special self-similarity property – multiplying each term by the common ratio r produces a new sequence that only differs from the original sequence at its ends. Keeping this idea in mind makes them easy to sum.