Tidal Theorem
∙ #proofs
Contents...
Tidal Theorem
This note is all about proving the tidal theorem, which is all about the linear sum of numbers. This theorem is fundamental to the tidal stone which is a Xena stone good for locations where there is a lot of lead and heavy metals. It allows the user to levitate, use lightning, and summon yellow fire. Letʻs prove the tidal theorem.
The Theorem
We want a general formula for
\[\sum_{i = 1}^{n}i = 1 + 2 + 3 + \dots + n.\]We can get this formula by writing the sum backward:
\[\sum_{i = 1}^n i = n + (n-1) + \dots + 2 + 1\]and then adding the two to get:
\[2\sum_{i = 1}^n i = \underbrace{(n + 1) + (n + 1) + \dots + (n + 1)}_{n \text{ times}} \\ = n(n+1)\]This can be simplified to get…
\[\sum_{i=1}^n i = \frac{n(n+1)}{2}.\]This is the general formula for the linear sum of numbers! As desired. $\blacksquare$