a and 1/a


math_equation_dice_d6In the last Carnival of Mathematics, two of the number facts were:

  • 1/49 is the sum of the series 0.02+0.0004+0.000008+…
  • 49 is the sum  the series 0.98+0.982+0.983+…

(Incidentally, I originally tried to put parentheses around the 0.98, but having an 8 and ) next to each other made the end format as 8) and that looked pretty funny.)

It turns out that it’s not a coincidence that the ratios in the two geometric series (0.02 and 0.98 ) add to 1.  As proof, suppose that:


The first term and ratio of this geometric series both equal x and so the series sums to \frac{x}{1-x}.  But we said this series was equal to  \frac{1}{a} so it follows that

a=\frac{1-x}{x}=\frac{(1-x)}{1-(1-x)}, which is the sum of the geometric series whose first term and ratio both equal (1-x).  In other words,

a = (1-x)+(1-x)^2+(1-x)^3...

Not the most exciting fact in the world, but still intriguing.

[As a further aside, if a is an integer then it turns out that x=\frac{1}{a+1}.  For example, \frac{1}{4}=(\frac{1}{5})+(\frac{1}{5})^2+(\frac{1}{5})^3... and therefore 4=(\frac{4}{5})+(\frac{4}{5})^2+(\frac{4}{5})^3....  I’m not sure if this makes the series more or less interesting, so I’ll pretend the answer is more.]

Blame Credit for this post actually goes to TwoPi, who first came up with the sums for 49 and 1/49 and who noticed the pattern of adding to 1.  Credit for the photo goes to Arjan Dice; it’s published here on Wikipedia.


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