# Parabolic Staking - Proof 1

# Proof that the boost multiplier approaches 2 as t approaches infinity

We define:
n - a given interval
t - the time duration of an interval in days
m_n - the multiplier at a given interval.
b_n - the boost at a given interval. This is the amount by which the multiplier will increase at each interval

According to our definition:
m_n - should increase at every interval, at a linearly decreasing rate, start at 1 and approach 2 as n approaches infinity.
b_n - should decrease at each interval and approach zero as n approaches infinity.

# Defining the Curve

a - The initial boost, equivalent to b_n. Describes the steepness the curve at the start.
r - Boost decay coefficient. Describes how aggressively the boost decreases over time in order for m_{max} to never surpass 2.

We therefore define the boost and multiplier as follows:
b_0 = a
b_{n+1} = b_n * r
m_{n+1} = m_n + b_n

# Proof of the Curve

We will proceed to prove that:
m_n \rightarrow 2 as n \rightarrow \infty

We know that b_0 = a and m_1 = 1 + b_0.

Since b_{n+1} = b_n \cdot r, we can express b_n as a \cdot r^n for n \geq 0.

The total multiplier after n intervals, m_n, is the sum of the initial multiplier (1) and the sum of boosts up to the n-th interval. Therefore,

 m_n = 1 + \sum_{i=0}^{n-1} b_i 

Substituting b_i with a \cdot r^i, we get:

 m_n = 1 + \sum_{i=0}^{n-1} a \cdot r^i 

This is a geometric series where the first term is a and the common ratio is r. The sum of the first n terms of a geometric series is given by:

 S_n = a \cdot \frac{1 - r^n}{1 - r} 

Substituting S_n into the expression for m_n, we get:

m_n = 1 + a \cdot \frac{1 - r^n}{1 - r} 

Since a + r = 1, we can substitute a with 1 - r, giving us:

m_n = 1 + (1 - r) \cdot \frac{1 - r^n}{1 - r} 
m_n = 1 + 1 - r^n 

As n \rightarrow \infty, r^n \rightarrow 0 (because 0 < r < 1 for a decaying series). Therefore, m_n approaches: m_n \rightarrow 1 + 1 - 0 m_n \rightarrow 2

Thus, it is proven that m_n \rightarrow 2 as n \rightarrow \infty, given the definition of our curve.