64 - Formula

[ Sn = a*r^(n-1) ]; n≠1,

Lynn wrote down the formula represented by the grid game using magic as ink. He then looked at the apprentices who were still recovering from the terror of exponential growth and spoke loudly.

"Why don't you all take note of this formula?"

Jenny and the others quickly flipped through the pages and used feather quills to transcribe the completely incomprehensible formula. Elok, who was standing there in a daze, was the best example of what happens when you don't study mathematics!

"Professor Lynn..." Elok looked at Lynn, stuttering and with tears welling up in his eyes, on the verge of crying.

"Elok, I just heard you say that you're good at mathematics?" Lynn asked.

"I'm not, I didn't, don't say that..."

Elok hastily shook his head, trying to deny it, but Lynn waved his hand and continued to speak.

"Here's the deal: if you can accurately calculate how much the thirty-six squares of copper coins add up to before tomorrow's class, then we'll call off this bet!"

"Of course, you only have one chance!"

Lynn glanced somewhat regretfully at the large pile of coins on the table, but he had no intention of taking them.

First of all, the other side couldn't come up with that much money. Secondly, using a game to empty the savings of the apprentices in a single go would be too intimidating.

Who would dare to attend his math class after that?

Elok stared at the floating magical fluorescent lights, completely unable to understand what these mathematical symbols were. He even began to doubt if he knew how to do math at all...

Previously, he had been envious of Elok, but now Pierce immediately became gloating, grateful that he had hesitated for a moment, or he would be the one making a fool of himself.

But Lynn's tone suddenly changed as he looked at everyone present and continued speaking.

"The same goes for all of you!"

"If I remember correctly when Elok borrowed money from you, he said, 'If he manages to fill all the squares, he'll repay double.'"

"Unfortunately, Elok didn't achieve that, so the promise is invalid. Not only won't you get more gold coins, but the money you bet earlier will remain with me!" Lynn reached out and picked up a few gold and silver coins from the table, saying with a hint of playfulness.

At these words, the classroom for the math class erupted in a chorus of wailing. Pierce and the others were burning with anger, kicking at Elok. This little guy was trying to cheat!

Lynn raised his voice a bit and pointed to the series summation formula floating in the air. "There is a simpler method for performing series addition. If it were me, I could calculate the sum of these numbers within ten seconds, provided I know the value of the last number!"

"The homework assignment I'm giving you is to find the pattern and solve using the formula I wrote! Whenever you figure it out, that's when you can take this money back!"

Lynn surveyed the students who were diligently calculating with feather quills in hand, nodding with satisfaction. Only this kind of firsthand experience could truly make them feel the charm of mathematics!

Apart from that, he was also using this game to assess the mathematical level of these apprentices.

From Elok's performance, it seemed that they had already mastered the most basic rules of arithmetic. Perhaps next time, he could start teaching slightly more challenging formulas!

The two-hour math class quickly came to an end.

Although they didn't learn any new magic, all the students maintained a high level of enthusiasm, driven by the idea of getting their deposits back. They eagerly shared their experiences and findings with each other in the classroom.

In just one day, the name of Professor Lynn's grid game had already spread.

To let more people understand the charm of exponential growth, Lynn directly moved the grid to the entrance of Iyeta Academy, changed the grid to seven rows and seven columns, with a total of forty-nine squares, and put up a sign with the rules of the bet written on it.

If anyone could fill these squares with the corresponding copper coins, he would give away the alchemical formula used to destroy Harbor Town.

Yes, it only required starting with one copper coin and filling the rest of the squares following the rule that each coin in the next square was twice the previous one. Even after filling, you could take the money back. He wouldn't take a single penny!

This had not only caught the attention of wizard apprentices but also some professors at the academy.

Was there really such a good deal?

While these alchemical formulas required a lot of preparation to unleash their powerful effects, they could prove incredibly effective in specific situations.

You see, the power of these formulas could rival even fifth or sixth-tier magic!

In the evening, while most of the students were leaving school or returning to the dormitories, the Professor of Morphology, Kevin, secretly approached the combined table to study the rules on the grid.

Did each square's number really need to be twice the previous one?

Kevin stroked his chin and pondered it internally. Just then, a familiar voice came from behind.

"No need to calculate; it's impossible to fill all these squares," said Professor Phillip.

"Do you know how much savings I have, Professor Phillip?" Kevin asked somewhat displeased. He had many friends, and these were just coins for filling squares. Once they were filled, he could take them back, or even borrow a bit more if needed.

Could these squares really be as impossible to fill as the rumors said, to the point of being unable to even use the entire wealth of Iyeta Harbor Town?

Kevin scoffed at the idea. This rumor was just too far-fetched!

"Do you know how many copper coins it would take to fill the last square of these seven rows and seven columns?" Phillip said somewhat helplessly. "It's 2.81 quadrillion!"

"2.81 quadrillion copper coins? That's quite a lot..." Kevin was somewhat surprised, but after a pause, he suddenly realized that Phillip had said... quadrillion?

"Wait, are you sure you didn't make a mistake?" Kevin was utterly bewildered. This was the first time he had heard of a quadrillion.

"Of course not! I've double-checked it myself..." Phillip said without much patience. He had been somewhat surprised when he first heard about it from the apprentices, and it had taken him nine full sheets of paper to calculate because the numbers were just too long.

Kevin mentally tried to calculate, and though it was just basic multiplication, the amounts grew increasingly immense as he went on. He couldn't even keep up with mental calculations after a certain point and had to resort to rough estimations.

Based on the experiences of the first few columns, it seemed like the sum increased by about tenfold every three to four squares.

With a total of forty-nine squares, that would be... at least a quadrillion times!

Kevin gasped for a breath; this kind of growth was terrifying!

Not to mention emptying Iyeta Harbor, it seemed impossible even to gather that much money across the entire continent!

Kevin was relieved that he had chosen to sneak out to try this when there were not many people around; otherwise, he would have made a fool of himself in front of the students.

In his heart, Kevin celebrated this choice and then noticed some strange characters engraved on the grid in front of him.

[ Sn = a*r^(n-1) ]

"What are these runes? Some kind of magical symbols?" Kevin asked in confusion.

"It seems to be... a geometric series summation formula? That's probably what it's called," Phillip said uncertainly. "Professor Lynn said that if you can calculate the final number, he can add up the copper coins in these squares in ten seconds."

"This is an immense amount of data; can it really be calculated in ten seconds?" Kevin furrowed his brow.

"Master Herlram seems to have already deciphered the meaning of this mathematical formula," Phillip sighed. "In just an afternoon, he's truly a great wizard."

"Did the master tell you what these mathematical symbols mean?" Kevin asked eagerly. After going through the exponential calculation, he found this so-called math rather intriguing.

"He didn't," Phillip regretfully shook his head. "Master Herlram only had me come to add a small reward to this game. Whoever figures out the meaning of this formula first can get a reward of twenty magical gold coins!"

As he spoke, Phillip took out these gold coins and magically affixed them to the table.

***

At this moment, the subject of discussion among the people was Professor Lynn, the new teacher of mystical mathematics, who was leisurely seated on the couch in his room. He was enjoying dinner brought by a fairy while simultaneously perusing the books he had borrowed from the academy's library: "Fundamental Principles of Magic" and "Analysis of Elemental Magic."

Although life as a professor at Iyeta Academy was exceedingly comfortable, Lynn remained vigilant. He never let complacency creep in. Everything in the land of wizards was still a mystery to him, and in case trouble arose, his strength was the only reliable companion.

Moreover, as a professor, he had to brush up on some fundamental magical knowledge. But the two books written by legendary wizards presented a formidable challenge. They were filled with perplexing terms like "Theros," "Enzethi," and "Coze," which left him utterly baffled. He could only painstakingly verify these terms based on the descriptions in the books.

It took him over three hours to decode the content, coupled with his modest understanding of magic since his arrival in this world. Eventually, he managed to grasp the first half of the book.

Firstly, his initial speculations were not far off the mark. Wizards indeed harnessed and manipulated elements to cast spells.

However, the abilities of regular wizards were quite limited. Not all elements could be controlled, and precision was lacking. For instance, they couldn't disintegrate molecules or atoms; they could only rearrange them in simpler combinations.

"Hydrogen, oxygen, nitrogen, phosphorus..." Lynn silently recalled the four elements he had manipulated. He couldn't help but make a few educated guesses.

First, the difficulty of element manipulation might be related to the periodic table. Official wizards might only be able to control short-lived elements up to the third row, while long-lived elements in the fourth to seventh rows might be beyond their reach.

Second, it could be based on the number of atomic nucleus charges.

Third, it could involve a division between metal and non-metal elements.

Lynn had conducted experiments, attempting to use Zero Tier Magic, "Elementary Material Deconstruction," to break down a steel longsword he held. The result was clear: he had failed. No matter how hard he tried, the sword remained unchanged.

This indicated that as a wizard apprentice or even as a regular wizard, he couldn't directly affect metallic elements.

Of course, things might differ for grand wizards and legendary wizards.

According to the Fundamental Principles of Magic, [Elementary Material Deconstruction] had two advanced spells:

Fourth tier spell, [Advanced Material Deconstruction,] and

Fifth tier spell, [Great Disintegration.]

Coincidentally, the fourth and fifth tiers marked the distinction between grand wizards and legendary wizards.

As Lynn pondered, he had 071 records of all these details in his research archive. Then, he turned to the second half of "Fundamental Principles of Magic," which delved into the mystical nature of magic.

This legendary wizard regarded magic as the essence of the entire universe, the most fundamental force that comprised all matter. It was the power of creation itself.

"Essence, huh?" Lynn furrowed his brow. According to his observations in recent days, magic appeared to permeate every corner of the world. However, unlike elements, he could sense it but couldn't truly "see" it. Only through magic such as "Magic Missile" could he momentarily manifest magic into a visible form.

Never mind; it seemed this was not something he could research with his current level of understanding.

Lynn massaged his somewhat aching temples and decided to set aside this issue for now. Instead, he opened "Analysis of Elemental Magic."

[Fireball], [Corrosion], [Thunderous Roar], [Flame Touch], [Minor - Fiery Grasp], [Explosive Flame], [Toxic Domain]...

The book contained descriptions of seventeen spells in total, ranging from first to third-tier magic. Lynn glanced at the so-called second and third-tier magic briefly but then focused his attention on first-tier magic.

With the aid of his intelligent assistant, he could perform magic in the capacity of a wizard apprentice, limited to learning and using First Tier Magic. Third Tier Magic, such as the principle behind "Toxic Domain," involved converting magic into some poisonous gas, possibly chlorine or fluorine, elements he couldn't easily acquire within a short time.

Being unable to decipher meant being unable to convert and use. His mental power was far from sufficient to cover such a vast range. In close quarters, it was better to simply adjust the oxygen content in the air, either increasing it or decreasing it, to achieve a similar effect.

Given his current capabilities, learning and refining First Tier Magic was the most practical approach.

Moreover, First Tier Magic was divided into common and advanced categories.

High-tier magic was essentially a simplified version of higher-level spells or an enhancement of lower-level spells. For instance, his "White Phosphorus - Fireball" technically belonged to high-tier First Tier Magic because of the lethality of white phosphorus, making its power akin to that of Second Tier Magic.

Another advanced spell, "White Phosphorus - [Minor - Fiery Grasp] possessed destructive capabilities approaching that of Third Tier Magic.

However, Lynn was well aware of the limitations of white phosphorus.

Despite its effectiveness against less-informed bishops, it was bound to lose its punch when dealing with wizards. After all, the two key characteristics of white phosphorus had possible countermeasures for official wizards. If the information were to leak out, achieving significant victories would become much more challenging.

So, he needed to acquire more magic to expand his arsenal.

TL/n - 

In a Geometric Sequence, each term is found by multiplying the previous term by a constant.

6x6 = 36 square 

1 2 4 8 16 32 64.... up to 36 times 

a = 1 (first term)

r = 2 (common ratio/multiplying constant) 

Formula => a*r^(n-1) 

= 1*2^(36-1) = 2^35 = 34,359,738,368

= 34 billion

***

You can calculate for the 7x7 = 49