I Find Blackboards Cool

Probability Level 4

I have an infinite number of 1s1's written on a blackboard.

Jake chooses 22 of the integers pp and qq and replaces them with p+q4=k1\dfrac{p+q}{4}={ k }_{ 1 }. (he removes pp and qq and then writes p+q4\dfrac{p+q}{4})

Now he repeats the process with the number k1{ k }_{ 1 } and another integer to achieve k2{ k }_{ 2 }, and repeats again with the number k2{ k }_{ 2 } and another integer to achieve k3{ k }_{ 3 }

k2=1+k14,k3=1+k24{ k }_{ 2 }=\frac { 1+{ k }_{ 1 } }{ 4 } ,\quad { k }_{ 3 }=\frac { 1+{ k }_{ 2 } }{ 4 }

Since he is an immortal, he does this again and again until he is left with only a single number...

Given that this number can be expressed as ab\frac{a}{b}, find a+ba+b


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