## Curve V1 Equation We'll cover the Curve V1 equation, which can be found in the Curve whitepaper. The general equation for *n* tokens is shown below. *n* represents the number of tokens, which can be greater than or equal to 2. ``` kai * sum all of the token balances from i = 1 to i = n + multiply all of the token balances from i = 1 to i = n = kai * (D + (D / n) raised to the power of n) ``` This equation is a combination of the constant sum and constant product. We can substitute the *kai* symbol with its corresponding terms, which is *A* multiplied by all of the token balances, divided by *D* / *n*, and then raised to the power of *n*. ``` A * sum all of the token balances from i = 1 to i = n / (D / n) raised to the power of n * sum all of the token balances from i = 1 to i = n + multiply all of the token balances from i = 1 to i = n = A * (D + (D / n) raised to the power of n) ``` The whitepaper also contains an alternative form of the Curve V1 equation. The form is shown below: ``` A * n raised to the power of n * sum all of the token balances from i = 1 to i = n + D = A * D * n raised to the power of n + D raised to the power of n + 1 / product of all coins from i = 1 to i = n ``` We can derive this alternative form from the general form by dividing both sides of the equation by *D* divided by the product of all the tokens. Let's simplify the left side of the equation first: ``` A * n raised to the power of n / (D / n) raised to the power of n * sum all of the token balances from i = 1 to i = n + D ``` *A* and the product of all of the tokens will cancel out on both the top and the bottom. *D* raised to the power of *n* will cancel out with *D* raised to the power of *n - 1* multiplied by *D*. This will leave us with: ``` A * n raised to the power of n * sum all of the token balances from i = 1 to i = n + D ``` Now, let's simplify the right side of the equation: ``` A * (D + (D / n) raised to the power of n) / (D / product of all coins from i = 1 to i = n) ``` The product of all the coins cancels out on both the top and the bottom. *D* raised to the power of *n* will cancel out with *D* raised to the power of *n - 1* multiplied by *D*. This will leave us with: ``` A * D * n raised to the power of n + D raised to the power of n + 1 / product of all coins from i = 1 to i = n ``` Putting the left and right side together, we get: ``` A * n raised to the power of n * sum all of the token balances from i = 1 to i = n + D = A * D * n raised to the power of n + D raised to the power of n + 1 / product of all coins from i = 1 to i = n ``` This is the alternative form of the Curve V1 equation.