create a formula for 1,1 2,2 3,3 4,4 5,5 6,6 7,7 8,8 9,9 10,10 if it takes 100 years to decay

Answer 1

It seems you are looking for a mathematical representation of the pairs (1,1), (2,2), ..., (10,10), and you want to incorporate a concept of decay over a period of 100 years.

Assuming you are looking for a decay function to model the values over time, we can use a standard exponential decay formula:

\[ N(t) = N_0 \cdot e^{-kt} \]

where:

• \( N(t) \) is the quantity at time \( t \),
• \( N_0 \) is the initial quantity,
• \( k \) is the decay constant,
• \( e \) is the base of the natural logarithm,
• \( t \) is time (in years).

If you want to represent the pairs (1,1), (2,2), ..., (10,10) as initial values and have them decay over 100 years, you can define each pair as follows:

1. For each pair (i, i), you can take \( N_0 = i \).
2. The decay constant \( k \) can be determined by the fact that after 100 years, the value should approach a certain value. For example, if you want the quantity to approach 0, you can set \( N(100) = 0 \).

However, since \( e^{-kt} \) only approaches 0 asymptotically, we can choose a value that represents a decay sampling at \( t = 100 \) years, for example \( N(100) = i \cdot e^{-100k} \).

Let’s take a specific example where we decide that the remaining value after 100 years for each \( i \) will be less than 1. We can choose \( k \) so that each value decays efficiently over that time span.

Steps to form your formula:

1. Choose a suitable decay constant \( k \). For instance, if you want the value to be less than 1 after 100 years:
   • Set \( N_0 = i \)
   • Choose \( k = -\frac{1}{100} \), so that \( N(100) \approx i \cdot e^{-1} \) (roughly \( i \cdot 0.3679 \)).

Thus, for each pair \( (i, i) \), the function becomes:

\[ N(t) = i \cdot e^{-\frac{t}{100}} \]

Example:

For each pair from 1 to 10:

• For \( i = 1 \): \( N(t) = 1 \cdot e^{-\frac{t}{100}} \)
• For \( i = 2 \): \( N(t) = 2 \cdot e^{-\frac{t}{100}} \)
• ...
• For \( i = 10 \): \( N(t) = 10 \cdot e^{-\frac{t}{100}} \)

You can use this formula to find the values at any time \( t \) through 100 years for each initial pair. Adjust the decay constant \( k \) as necessary for different rates of decay.