Summing a Distribution to Find a Constant

The one thing that all probability distributions have in common is that all probabilities add to 1.
Suppose a random variable  
  follows the distribution  
\[P(X=x)=a (\frac{2}{5})^x\]
, where  
  is an unknown constant. We can find  
  using the condition  
\[\sum_x P(X=x)=1\]
\[\sum_{x=0}^{\infty} a(\frac{2}{5})^x=1\]
The sequence  
\[P(X=0)=a, \: P(X=1)=\frac{2}{5}a, ..., P(X=k)=(\frac{2}{5})^k a\]
  is a geometric sequence with first term  
  and common ration  
  so we can use the formula for the sum of a geometric sequence  
\[1=\frac{a}{1-2/5} =\frac{a}{3/5} \rightarrow a= \frac{3}{5}\]

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