When q≠1
Sn=a1(1-q^n)/(1-q) Lin En used magic as ink to write out the formula represented by the grid game, and then looked at the apprentices who had not yet recovered from the horror of exponential growth, and spoke loudly.
"Why don't you write this formula down?"
Johnny and the others quickly turned the pages, picked up the quills and copied down the formulas that they couldn't understand at all. Alok, who was standing there in a daze, was the best example of the consequences of not learning Mathematical Olympiad! "Professor Lin En..." Alok looked at Lin En, trembling and shouting, his eyes filled with tears, almost crying.
"Elok, I just heard you say that you are good at math?" Lin En asked.
I am not, I didn't, don't say anything...
Alok hurriedly shook his head and wanted to deny it, but Lin En waved his hand and continued speaking.
"How about this, as long as you can accurately calculate the total value of the copper coins in these thirty-six squares before class tomorrow, then this bet will be settled!"
"Of course, you only have one chance to speak!"
Lin En glanced at the large amount of coins piled on the square table with a hint of regret, but he had no intention of accepting them.
First of all, it was impossible for the other party to come up with such money. Secondly, it was a bit too scary that he let all the apprentices spend all their savings just by playing a game.
Who would dare to attend his Mathematical Olympiad class in the future? Alok stared blankly at the floating magic fluorescent lights, not understanding what these operation symbols were. He even began to doubt whether he could do arithmetic...
Pierce, who had been envious of Elroq, immediately became gloating. Fortunately, he hesitated for a moment, otherwise he would be the one making a fool of himself.
Unexpectedly, Lin En changed the subject, looked at the people present, and continued speaking.
"You too!"
"If I remember correctly, when Alok borrowed money from you, he said, 'If he can fill all the squares, he will pay you back double.'"
"Unfortunately, Alok failed to do this, so the promise is invalid. Not only will you not get more gold coins, but the money you deposited before will have to continue to be kept with me!" Lin En reached out and picked up a few gold, silver and copper coins on the square table, and said playfully.
Upon hearing this, there was a sudden wailing in the Olympiad Mathematics classroom. Pierce and others kicked Alok in anger. This kid actually cheated!
Lin En raised his voice a little and pointed at the exponential summation formula floating in the air. "There is a simple way to add exponential operations. If it were me, I could calculate the sum of them in ten seconds if I knew the last number!"
"The after-class assignment I give you is to find the pattern and solve the formula I wrote! When you get the answer, you can take the money back!"
Lin En looked around at the students who were holding quills and calculating hard, and nodded with satisfaction. Only this kind of personal experience can make people truly feel the charm of the Mathematical Olympiad! In addition, he also used this game to find out the mathematical level of these apprentices.
Judging from Alok’s performance, they have already learned the most basic calculation rules. Maybe we can start by teaching them slightly more difficult formulas next time!
The two-hour Olympiad mathematics class ended quickly.
Although they did not learn any new magic, all the students maintained great enthusiasm due to the desire to get their deposits back, and they even couldn't wait to share what they saw and heard in class with others.
In just one day, the name of Professor Lin En's square board game spread.
In order to let more people understand the charm of exponential growth, Lin En moved the square plate directly to the front door of Yieta College, and changed the squares into seven rows and seven verticals, a total of forty-nine squares, and erected a sign on it, rewriting the rules of the bet.
If anyone can fill up these squares with the corresponding copper coins, he will give away the alchemical formula used to destroy the harbor town.
That's right, you only need to start with one copper coin and fill it up according to the rule that the copper coins in the next grid are twice as much as the previous one. After filling it up, you can even take it back, and he doesn't charge a penny! This time, not only the wizard apprentices, but also several professors in the academy were somewhat tempted.
Is there such a good thing? This alchemical formula requires a lot of preparation to unleash its powerful power. Although it seems troublesome, it can bring out special effects in some special occasions.
You know, the power of this thing is comparable to five-ring or even six-ring magic! In the evening, while the students were leaving school or returning to the dormitory, Professor Kevin of Plastics sneaked to the merged square table and studied the rules on the square plate.
Does each square have to double the number in the previous one?
Kevin touched his chin and pondered in his mind. Just as he was thinking, a familiar voice came from behind.
"Don't bother calculating. It's absolutely impossible to fill up all these squares!"
"Do you know how much savings I have, Professor Philip?" Kevin asked with a little dissatisfaction. Besides, he knew a lot of good friends. Anyway, the money was just used to fill in the boxes and he could take it back after filling them. At worst, he could borrow some more.
Could it be that these squares are really as rumored, that even if all the wealth of the entire Yieta Harbor was put into them, they still couldn't be filled? Kevin scoffed at this statement. This rumor was simply too outrageous!
"Do you know how many copper coins it takes to fill in the last square of these seven rows and seven columns?" Philip said helplessly. "It's two hundred and eighty-one trillion!"
"Two million, eight hundred and one thousand copper coins, that's a lot..." Kevin was somewhat surprised, but after a pause, he suddenly realized that the unit Philip mentioned seemed to be... ten thousand... billion?
"Wait, are you sure you didn't make a mistake?" Kevin was completely confused.
It was the first time he heard of the unit of trillion.
"Of course not! I have verified it myself..." Philip said unhappily. He was also a little surprised when he heard the apprentices talk about it before, and then he used nine sheets of manuscript paper to finish the calculation because the numbers were too long.
Kevin silently did the mental arithmetic in his head. Although it was just the simplest multiplication, the numbers became larger and larger as he continued to calculate. It became difficult for him to sustain the calculation just by mental arithmetic, so he could only make a rough estimate.
Based on the experience of the previous columns, the amount will increase about tenfold every three to four grids.
There are a total of 49 grids, which means... at least one hundred trillion times! Kevin couldn't help but take a breath of cold air. Such growth is really too terrifying! Not to mention the Yieta Harbor, even if the entire continent was emptied, it would be impossible to raise so much money!
Fortunately, he chose to try it secretly when there were not many people around, otherwise he would have made a fool of himself in front of the students.
Kevin secretly rejoiced in his heart, and then he found that in addition to the rows of squares on the square table in front of him, there were also some strange characters engraved on it.
When q≠1, Sn=a1(1-q^n)/(1-q) "What are these runes? Some kind of magic symbols?" Kevin asked puzzled.
"This seems to be... the formula for summing a geometric series? That should be what it's called, right?" Philip said uncertainly. "Professor Lin En said that as long as he figures out the last number, he can figure out the total of the copper coins in these squares within ten seconds."
"Can such a large amount of data really be calculated within ten seconds?" Kevin frowned.
"Did the master tell you what these Olympiad symbols mean?" Kevin asked eagerly. After doing some exponential calculations, he found that this so-called Olympiad mathematics seemed quite interesting.
"No..." Philip shook his head regretfully. "Master Helaram only asked me to come here to add a small reward to this game. Whoever can first figure out the meaning of this formula will receive a reward of 20 magic gold coins!"
As he spoke, Philip took out the gold coins and used magic to fix them on the square table.
…
At this moment, the subject of everyone's conversation, Lin En, the new professor of Olympiad Mathematics, was sitting leisurely on the sofa in the room, eating the dinner brought by the goblins while checking the "Basic Principles of Magic" and "Analysis of Elemental Magic" borrowed from the academy library.
Although the life of a professor at Yieta College is very comfortable and he does not feel the slightest bit of crisis, Lynn will not slack off because of this.
Everything in the wizarding land was unfamiliar to him. Once he encountered any trouble, the only thing he could rely on was his own strength.
In addition, as a professor, he also needs to make up for some basic magic knowledge.
However, these two books on magic principles written by legendary wizards gave Lin En a headache, because the books were filled with a large number of words such as [Serlu], [Enzesi], [Koze], etc. that made him completely confused. He could only verify them bit by bit based on the descriptions of these nouns in the books.
It took more than three hours of analysis, combined with his slight understanding of magic after the time travel, Lin En could barely understand the first half of the book.
First of all, his previous guess was not wrong. Wizards did release magic by manipulating, influencing, and mimicking elements.
However, the abilities of formal wizards are very limited. They cannot control all elements, and their precision is not enough. At least, they cannot smash molecules or atoms. They can only perform simple arrangements and combinations.
"Hydrogen, oxygen, nitrogen, phosphorus..." Lin En silently thought about the four elements he had controlled, and couldn't help but have some guesses in his mind.
First, the difficulty of element manipulation may be related to the element period. For example, official wizards can only manipulate the short-term elements in the first three rows of the periodic table, but cannot manipulate the long-term elements in the fourth to seventh rows.
The second is determined by the number of nuclear charges in the element.
The third is to divide it by metal and non-metal elements.
Lin En had experimented with this before, trying to cast zero-ring magic [Elementary Matter Deconstruction] to disintegrate the steel sword in his hand.
It was obvious that he had failed! No matter how hard Lin En tried, the sword remained motionless.
This means that a wizard apprentice or even a formal wizard cannot directly affect the metal element with his abilities.
Of course, this is not necessarily the case for great wizards and legendary wizards.
Because according to the "Basic Principles of Magic", [Basic Material Deconstruction] also has two advanced magics.
They are the fourth-ring magic [Advanced Material Deconstruction] and the seventh-ring magic [Great Disintegration]! Coincidentally, the fourth and seventh rings are the dividing line between being promoted to a great wizard and a legendary wizard! While thinking, Lin En asked 071 to record these contents in the research file, and then flipped through the second half of "Basic Principles of Magic", which described the various magic powers even more mysteriously.
This legendary wizard regards magic as the origin of the entire universe, the most basic substance that constitutes all matter, and the true power of creation!
"The origin?" Lin En frowned. Based on his observations over the past few days, magic power seemed to exist in every corner of this world. However, unlike elements, he could only sense it but could not truly 'see' it. Unless he used magic such as [Magic Missile], the magic power could be materialized for a short period of time.
Forget it, this is obviously not something he can study clearly at his current level...
Lin En touched his head which was aching from thinking too much and decided to put this question aside for the time being. Then he opened "Detailed Explanation of Elemental Magic".
[Fireball], [Corrosion], [Loud Spell], [Touch of Ice], [Secondary - Hand of the Fire Demon], [Pyroblast], [Poison Field]...
The entire book "Analysis of Elemental Magic" records a total of seventeen types of magic from the first ring to the third ring.
Lin En glanced at the so-called second and third ring magics, and then stopped paying attention to them. Instead, he turned his attention to the first ring magics.
With the help of the intelligent brain, although he can complete magical mimicry as a wizard apprentice, which is something only formal wizards can do, he is only limited to learning and using first-level magic.
For example, the principle of the three-ring magic poison field is to use magic power to transform it into some kind of toxic gas, which could be chlorine or fluorine. In short, these are not elements that he can get in a short time.
If it cannot be analyzed, it cannot be converted and used. His mental power is far from enough to cover such a large area.
In close-range situations, it is better to simply lower or increase the oxygen content in the air, which can achieve a similar effect.
With his current abilities, learning and improving the first ring of magic would be the least troublesome thing to do.
And first-ring magic is also divided into ordinary and superior levels.
The so-called higher-level magic is actually a simplification of higher-level magic, or an enhancement of lower-level magic.
For example, his [White Phosphorus - Fireball], strictly speaking, belongs to the first-level upper magic, but due to the highly toxic and difficult-to-destroy nature of white phosphorus, its power reaches the level of second-level magic.
Another improved magic [White Phosphorus - Hand of the Fire Demon] has a destructive power close to that of a third-ring magic.
But Lin En was well aware of the limitations of white phosphorus.
Even though he used this thing to kill people everywhere in the Sekas Empire, he even killed Anluk, who was comparable to a three-ring wizard.
Lin En knew very well in his heart that this thing was fine for bullying those uneducated bishops, but its power would be greatly reduced if it was used against wizards.
After all, formal wizards can deal with the two important characteristics of white phosphorus. Once the information is leaked, it will not be easy to achieve good results.
Therefore, he must learn more magic to increase his trump cards...
(PS: I ask for anything for the new book! Today, two chapters will be released together, and you can also discuss whether it is better to have more things of this scientific principle nature, or to keep it as brief as possible.) Thanks to book friend 20200302113752551 for the reward of 1500 Qidian coins, Xi Xi Xi Xi and That Moment's Smile is Like a Flower 2020 for the reward of 500 Qidian coins, and Mo Yuyang, Ten Thousand Years of Tianshan Snow Lotus, and Local Planning Bureau for the reward of 100 Qidian coins.
(End of this chapter)