#
#
Parabolic staking proof

###
#
Mathematical Proof of Parabolic Reward Mechanism

####
#
Premises

Given the following premises: ( a + r = 1 ) ( b_0 = a ) ( b_{n+1} = b_n \times r ) ( x_{n+1} = x_n + b_n )

####
#
We will prove that:

( x_n \rightarrow 2 \times x_0 ) as ( n \rightarrow \infty )

####
#
Step 1:

Solve for ( x ) in terms of ( a ) and ( b ) From equation 1, we express ( x ) as:

##
#
Step 2: Express ( x ) in Closed Form

We can express ( x ) in a closed form as it forms a geometric sequence:

##
#
Step 3: Sum of ( x ) as ( n ) approaches infinity

The sum ( S ) of an infinite geometric series with ratio ( r ) is: So in this case, ( S = \frac{1 - r} ).

- ( a + r = 1 )
- ( b_n = 0 )
- ( a )
- ( b_{n+1} = b_n \times r )
- ( x_{n+1} = x_n + b_n )

##
#
Step 4: Represent ( x ) as a Sum

Using the given recursive equation ( x_n = x_{n+1} + b_n ), we write: Since ( x_0 ) is given as an initial value and ( S = \frac{1} ), ( x_n ) will be:

##
#
Step 5: Limit as ( n \rightarrow \infty )

The value of ( x_n ) becomes constant after the first term, since ( b_n = 0 ): Thus, ( x_n ) approaches ( 2 \times x_0 ) as ( n ) approaches infinity.