I was a top student in America.

"Ryan, did you solve this recurrence sequence problem? It seems really complicated."

"I've done it. For this problem, you just need to calculate the eigenvalues."

"I see. I'll give it a try, but I'm afraid the calculations will be too complex."

"Just calculate it by brute force, otherwise you can't deduce the general term. The middle steps of this problem are really disgusting, too easy to make mistakes, but it's not difficult."

……

In a classroom at Whitney Young High School, some time before the workshop was to begin.

Most of the students had arrived early and were gathered in twos and threes, checking their thinking against the pre-distributed study materials.

As Li Aogang walked outside, he heard these complaints and immediately matched them with the questions in his handouts.

They were discussing the last major question in Brown's lecture notes.

This morning at home, he also got stuck on the complicated steps of this problem for a while.

It wasn't until I was sitting on the swaying red line train just now that I came up with a simple solution:

In fact, as long as you subtract the recurrence relation and construct a geometric sequence, the three-line equation can be solved.

He went over the steps in his mind again, then pushed open the door and walked into the classroom.

Upon hearing the door open, the boy closest to the door stopped writing and glanced at the unfamiliar face.

The boy, wearing a crisp light-colored Oxford shirt, adjusted his glasses and casually asked:

Who are you looking for?

"I'm here to attend the workshop."

"Which school? Linkin Park High School or Lane Polytechnic High School?"

Wyndham High School.

"what?"

The bespectacled boy paused for a moment, and upon hearing that it wasn't a famous school, he immediately lost interest in asking further questions. He simply said "Oh," and turned back to continue studying the calculation steps for the eigenvalues ​​with his deskmate.

The others in the classroom only glanced at it casually before going about their own business.

Li Aole was happy to have some peace and quiet, so he went to the back corner, picked an empty seat, and casually tossed his bag onto the table.

He had just opened his handout and written two lines of equations when the front door was suddenly pushed open.

"Alright, everyone find a place to sit down."

A middle-aged man with a severely receding hairline strode in, clutching a stack of lecture notes in his hand.

The previously noisy classroom immediately fell silent.

Mr. Green glanced around, his gaze lingering briefly on Li Ao, a stranger in the back row, before moving away.

Without a word, the competition coach turned around, grabbed the chalk, and quickly wrote a problem on the blackboard:

The sequence {a_n} satisfies a_1 = 1, a_2 = 5, and a_{n+2} = 4a_{n+1} - 3a_n. Find a_20.

"Who solved this last recursive problem?" Green asked, dusting off his hands with chalk dust.

"I'll do it." The man in the shirt who had just asked Li Ao about the school calmly closed his notebook, raised his hand, and then pushed back his chair and stood up.

"Oh, it's Ryan. Come on up and give it a try." Mr. Green nodded.

Ryan went up to the podium, grabbed a piece of chalk and started writing, clearly having already prepared a draft.

His approach was extremely straightforward: first, he set up the characteristic equation x^2 - 4x + 3 = 0 and solved for the characteristic roots 1 and 3; then, he set the general term a_n = A + B * 3^n and substituted it into the initial conditions to solve the system of equations to find the coefficients.

Before long, the blackboard was covered with a large number of complicated calculation formulas.

Green stood to the side with his arms crossed, watching quietly. After Ryan finished writing, he suddenly asked, "Why didn't Matthew come today?"

"He went to participate in the state physics competition," someone below replied.

Green raised an eyebrow and turned to look at Ryan.

Ryan has just calculated the general term a_n = 2 * 3^{n-1} - 1, and is about to substitute n=20 into it.

"Alright, Ryan, that's enough for the general terms."

Green stopped him, then looked at the dozen or so students present, "Besides this rote calculation approach, is there a more efficient solution?"

A moment of silence fell over the classroom.

The "eigenvalue method" is a standard solution for recursive sequences. Although it is cumbersome, it is robust and can be applied to most types of problems.

"Isn't this the correct solution to the problem?"

Ryan muttered something under his breath, threw the chalk back into the trough, and went back to his seat with his head down.

Green's gaze swept past the evasive eyes in the front row, finally crossing half the classroom and landing in the back corner:

"Your name is Leo, right?"

Several students who were nearby turned around and looked at Li Ao, who was still doing practice problems. The others also glanced at him in the same direction as Green.

"It takes more than an hour to get here from the South District by taking the Red Line train, right?"

Green's tone wasn't as harsh as before; instead, it carried a hint of encouragement. "You've come all this way, don't just sit back and do nothing. Come up and give it a try. It's okay if you can't write down the answer; just write down some ideas so I can get a feel for you."

As soon as these words were spoken, a murmur rippled through the classroom.

"From the South District?"

"Why would someone travel so far to attend a competition class?"

Li Ao didn't say anything, but closed his pen, stood up, and walked onto the podium amidst a dozen curious and even scrutinizing gazes.

He picked up half a piece of chalk from the chalk box, but didn't touch Ryan's derivation process that took up half of the blackboard. Instead, he found an empty space next to it.

The first step is to make a difference: a_{n+2}- a_{n+1}= 3(a_{n+1}- a_n)

The second step is to let d_n = a_{n+1}- a_n, then {d_n} is a geometric sequence with the first term 4 and the common ratio 3.

The third step is to accumulate: a_n = 1 + 4 * (3^{n-1} - 1) / (3 - 1) = 2 * 3^{n-1} - 1

There are only three lines of formula in total.

After finishing writing, he nodded to Mr. Green, then threw the chalk into the trough, clapped his hands, and turned to leave the stage.

"Holy crap... we can just construct a geometric sequence by calculating the differences?"

"Damn, yeah, doing it this way? The constant is completely eliminated..."

"Three lines... and you're done writing?"

The classroom's silence was completely shattered, and someone subconsciously started scribbling on their draft paper.

Ryan stared at the three lines of equations, and the ballpoint pen he was twirling in his hand fell to the table with a "thud".

You spend ages painstakingly setting up equations and finding the undetermined coefficients, while someone else just casually does a subtraction?

The contrast is too stark.

Green stared at the blackboard for two seconds, a rare smile appearing on his usually stern face.

He tapped his knuckles twice beside the simple set of steps, then nodded to Li Ao, who was walking back.

Where did you see this solution?

"I borrowed an old book called 'AMC Questions and Answers' from the public library on 61st Street."

Li Ao stopped next to his seat. "There's a number theory example problem inside, which uses constructed sequences to make an approximation proof. I borrowed some of its ideas."

"Actually, I initially tried to solve this problem using the characteristic root method, but I was bored on the Red Line train just now and had a bit of inspiration, so I changed my approach."

Green didn't ask any further questions.

The ability to grasp the underlying principles from the construction method of a number theory problem and apply them to a completely unrelated recursive sequence problem...

It seems Brown wasn't exaggerating on the phone; this kid did manage to catch up on 11th-grade algebra in just half a month through self-study, he's got some skills.

He looked away and picked up the chalk again: "Okay, let's look at the next question."

The rest of the morning was filled with intensive lectures. Green pushed the blackboard forward at an extremely fast pace, one problem after another, completely disregarding the well-being of those below him.

The cruelty of math competitions lies here: it's a game prepared only for a few geniuses, and it's never a fair and equal opportunity for everyone.

Li Ao remained huddled in the last row, his pen scribbling incessantly on the draft paper.

He discovered that as his intelligence attribute increased, the ability to apply knowledge to new situations also became stronger.

He could often connect the advanced techniques Green taught to a problem he had done before as soon as he heard the beginning, and his reaction speed was much faster than when he was doing practice problems.

"Having a coach makes all the difference..."

Li Ao thought to himself, no wonder all the rich kids he mentored in his previous life had to spend a lot of money to hire competition coaches.

Having someone summarize and analyze various question types is much more efficient than blindly doing practice problems on your own.

This one-hour class is equivalent to several days of hard study.

After a short lunch break, there was a two-hour acceptance test at 1 p.m.

The test paper had a lot of questions, and the classroom was so quiet that only the scratching sound of pens poking at the paper could be heard. No one dared to daydream.

Li Ao, however, leaned back in his chair, twirling his pen and methodically working through the equations.

"There's only half an hour left, let's hurry."

Green leaned back in his chair behind the podium, unscrewed the cap on his Coke, took a swig, and spoke up to remind him.

As soon as he finished speaking, a soft sound came from the back row of chairs scraping against the floor.

Li Ao carried the test papers to the teacher's desk:

"Mr. Green, I've finished. Can I hand in my paper now?"

Green paused for a moment, then glanced instinctively at the clock on the wall: "All done?"

Li Ao nodded.

"There's still half an hour left, shouldn't we check it again?"

"No, teacher. I have to rush back to work in a bit. My boss didn't approve my leave today, and I'll get docked pay if I'm late."

……