Let x = 4.16666...

10x = 41.66666...

Subtracting 1 from 2,

9x = 37.5

So, x = 75/18

Hence, any recurring fraction can be put in the form of a fraction.

What do you mean by form of ar^n ?

For your second fraction,

Let x = .4511414...

100 x = 45.11414...

So, 99x = 44.663

x = 44663/99000

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Harish