The school bell rings to mark the beginning of a school week. Apple Bloom, Scootaloo and Sweetie Belle all slowly canter their way into the class. They each take their seats in their own desks. Cheerilee walks into the class with a mysterious box and sets it down next to the class. She has a tremoring hoof since she has not taught this subject for 4 years.
“Ooh! What is that you’re holding, Ms. Cheerilee?” Sweetie Belle points at the box with a questionable attitude.
“I don’t know, but I will open it up later this day.”
“Okay class! It’s time for me to do this… Since I have taught you all the essentials in mathematics, let’s start with a mind-bender!”
The students groan as then get a piece of paper and place it on their desk and a pencil in their mouth.
Cheerilee then writes the following on the board:
1 + 2 + 3 + … + 98 + 99 + 100 =?
“5050!" Scootaloo declares.
“Yes, Scootaloo, that is the answer, and now explain how did you get the answer.”
“Well… Twilight kind of told me the answer to this problem…”
“Try solving it,” Cheerilee spits the shortened chalk onto the table. Apple Bloom raises her hoof.
“Yes?” Cheerilee requests her to speak.
“How do I do this? I don’t know how to add dots and numbers together.”
“Oh! I forgot to explain. The dots symbolize a continuation in the numbers, so If I write:”
1+2+3+…+8+9+10 = 55
“…Then the dots would resemble all the numbers that are in-between 3 and 8, so if I would write down the equation without the dots, it would look like this:”
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
“That makes it more difficult!” Sweetie Belle squeaks, thinking it as absurd.
“That is how mathematicians do it. They use it to eliminate the hassle of writing all the numbers in the middle,” Cheerilee explains, and as a result, the class gets back to work.
Scootaloo is then curious, and raises her hoof.
“So do they only write it this way?”
“No. They have another way to write it, and is quite simple and complex at the same time,” Cheerilee says and then grabs the chalk.
“Let’s see," Scootaloo stares at the board.
“No! Scootaloo, this is enough torture for me today!” Diamond Tiara objects from the back of the classroom.
Cheerilee writes the following on the board:
“Okay! Ms. Cheerilee! This is already giving me a headache!” Diamond Tiara exclaims, and Cheerilee laughs a little.
“I’ll show you the solution later.”
“So, what is that scary-looking E stand for?” Scootaloo points out.
“It is the Greek letter Sigma, it denotes the sum. So in this diagram, it says to add the number from 1 to 100 the old fashioned way since the expression the problem wants us to plug in is just n and add the next number you plug in.”
“What if the expression thingy is not just only one n?”
“Well, let’s say…” Cheerilee writes the following:
“So the n is the only thing that you plug the numbers in, and then you add the next?”
“You are on track, Scootaloo.”
Scootaloo whispers a quiet "Yes!" to herself as she turns around to Apple Bloom and says "You see? Twilight Times does pay off!" before she resumes looking at the chalkboard.
“Wow, math is sure complex,” Scootaloo seems intrigued.
“It is a complex subject. All of our underlying sciences all depend on math. I would say that math is the “engine” that powers all of the sciences.”
…
“Is everypony done?”
“No… This is too much work! I am still adding 57 right now…” Sweetie Belle declares.
“Well, there is a trick to this,” The whole class stops and looks at Cheerilee. Cheerilee grabs the eraser and erases the sigma notations, and works on the problem.
“A human being named Gauss worked on this problem when he was a first grader, and he found a way to solve this problem,” Cheerilee introduces the synopsis.
1 + 2 + 3 + … + 98 + 99 + 100 =?
“Now, how many pairs are there in this set of numbers between 1 and 100?”
“Let’s see… 100 divided by 2… 50!” Apple Bloom utters the answer.
“Good! So 50 pairs, and now, I will show you how the pairs come in,” She grabs the chalk, and circles the two ends of the equation.
1 + 2 + 3 + … + 98 + 99 + 100 =?
“What is 100 + 1?”
“101.” The class shouts out.
“Correct, so now we know 100+1 is 101, now…”
1 + 2 + 3 + … + 98 + 99 + 100 =?
100+1 = 101
99+2 = ?
“What is 99+2?”
“It’s still 101.” Sweetie Belle shouts out.
“What about 98+3?”
“101,” Scootaloo answered as the rest of the class look dazed.
“So, we can look and see that the pairs of the corresponding beginning and end terms are both all 101. If we would go further, 97+4 is 101, 96+5 is also 101, and 95+6 is 101!” Cheerilee raises her voice at the end, giving her a sense of enthusiasm.
“So, we have 50 pairs, and each pair sums to 101, sounds like we got some multiplication to do.”
50 * 101
“Our answer is, Scootaloo?”
“5050!”
50 * 101 = 5050
“…And that how Gauss did it,” Cheerilee concludes the conjecture.
“Wow. That is so much easier than adding it all up…” Sweetie Belle looks at her paper and up to the board.
“So, what I have introduced to you is a series. A series is a set of numbers with a same common difference and you add them all up. The following is an example of a series.” Cheerilee writes the following:
1 + 3 + 5 + … + 97 + 99 +101
“Notice the next number is two greater than the one preceding it, so therefore the common difference is adding 2,” Cheerilee pauses as the class jots down what Cheerilee just said.
“For these kinds of series, we call it an arithmetic series, which deals with adding the common difference.”
“So, there is a formula that ties this entire bad colt up…”
“In the first problem, we asked you fillies and colts to sum up the first 100 integers, so I am going to denote it as:”
“I then asked you about how many pairs are there, so we have:”
“And now, what are we missing in this equation?”
“The first and last terms!” Apple Bloom enthusiastically shouts out the answer.
“Good job! So we then multiply the number of the pairs by the sums of each pair:”
“You should get 5050 by doing this operation,” Cheerilee concludes.
“Ugh… Ms. Cheerilee? Where are our abaci? We can’t handle doing these large operators!” Scootaloo complained.
“Oh… Gosh… I forgot to bring the class supply…” Cheerilee grunts and says with a very tired and exasperated tone.
“I’ll be back,” Cheerilee runs out of the class. When she is unavailable, Diamond Tiara hops onto the ground, and walks to the teacher’s desk.
“What are you doing?” Scootaloo scowls at Diamond Tiara.
“Oh, nothing,” Diamond Tiara acerbically talks back, and then gets ahold of the box.
“Amazon…” Diamond Tiara reads, “Is that a country?” Silver Spoon walks up and accompanies Diamond Tiara.
“No! What are you two doing? You can’t reveal what Cheerilee has! What it the contents inside the box is a large secret?” Apple Bloom stood as the two naughty fillies grab the scissors. Before Apple Bloom and Scootaloo can get ahold of the scissors, it is already too late since the sealing tape is cut right down the middle.
“Ha! Now we can see what she is truly hiding!” Diamond Tiara flips the flaps away, and digs under the packing peanuts and the inflatable bags and they unveil something that they have never seen...
The object that lies in the box is a pseudo-rectangular object with a screen, and myraids of keys with numbers and symbols on it.There is a full QWERTY keyboard, and a large screen; on the left of the screen are function keys, and on the right are the operators.
“Texas Instruments TI-92… What?” Silver Spoon looks at Diamond Tiara, they both shake their heads in question.
“Hey, you three. Tell me what is this,” Diamond Tiara faces the CMC and Sweetie Belle walks up and investigates.
“I have never seen anything like this,” Sweetie Belle presses a button, and it does not work since the screen currently does not display anything.
“What’s a Texas Instruments? Is it something you play at concerts?” Apple Bloom questions.
“I really don’t know what it is…” Sweetie Belle says.
Scootaloo hears a stomping sound from the distance, and as she peers out the door she sees Ms. Cheerilee, running towards the school with saddle bags of abaci.
"Quick! Everypony get back!" The fillies quickly scurry back to their seats just in time when Cheerilee arrives.
"Okay class! Let's think of the formula we just wrote in a symbolic way," Cheerilee grabs the chalk and writes the following.
"Now, before you freak out, this is nothing special. It is all algebra. The n means how many numbers are there or how many numbers are you adding up. The a sub 1 means the first term, and the a sub n means the last term, hence the n, denoting the total number of terms. The previous example, 100, is the a sub n, which in this case, is a sub 100."
"Ooh..." Scootaloo realizes the reality of algebra as she starts writing down the formula.
"Now, you may panic," Cheerilee places both of her hoof on each of her two ears.
"AAAAAAAHHHHH! MY HEAD HURTS!" Silver Spoon bursts out in panic.
The glass gives her a moment to let her relax, and Cheerilee hands out the abacus to every student in this process. She resumes to the board, and starts speaking.
"Okay Class! Now, I will give you an arithmetic series. You will find something missing that I need to give to you."
20 + 24 +28 + ... +64 + 68 + 72 = ?
"Oh no! It's going by 4's! It's too much!" Apple Bloom exclaims. The class thinks altogether, and eventually, Scootaloo notices something.
"How many terms are in the series?" Scootaloo gives the answer.
"That is what we need to find!" Cheerilee writes something on the board.
"Do you remember last class where I introduced the arithmetic sequence formulas?"
"Oh! That formula... It's been a weekend, and I have forgotten it already," Sweetie Bell says.
"I sort of remember..." Scootaloo says.
"So, now we know that we need to find how many terms are in this sequence, what is the variable in question?" Cheerilee explains to the class.
"The N!" Apple Bloom exclaims.
"Very well done. It seems that you are quite proactive today since I have seen you slouching over for the past few weeks..." Cheerilee compliments.
"Now, make sure you follow me," Cheerilee resumes to the board and starts to manipulate the variables while Silver Spoon sits in the back and spaces out.
"First, let's move the A sub 1 to the other side by doing subtraction on both sides. You can see that the A sub 1 on the target variable side just cancel out due to opposite signs."
Cheerilee waits as the class writes down the description and process; once everypony is done, she continues on writing and explaining.
"Now, on the side with the N in it, we have two products: an N minus 1 bounded in parenthesis, and a D. We need to get rid of the D, and since it is a product, divide it on both sides so the D's cancel on the target side."
She pauses for the class the catch up.
"Now, we can get rid of the parenthesis for the N-1."
"Ms Cheerilee? Why does the parenthesis go away?" Scootaloo questions.
"It is now a standalone term since there is nothing to multiply by."
"Now, we need to move the one over, and the 1 has a subtraction sign in front of it. We need to add it to both sides to it would cancel on one side. Now, we can solve for N since it is all alone."
"So, we now have to plug in the information. You already have the variable designations from the previous lecture, so I'll let you work on your own."
... (5 minutes later)
"What did you get?" Scootaloo asks the members of the CMC.
"14 terms," The both speaks in unison.
"Yeah."
Meanwhile, in the back...
"This is too much for me! I still liked the day where we did 1+1... This is Twilight math!" Silver Spoon complains in a whisper.
"Well, just sleep through! She never notices!" Diamond Tiara whispers a solution.
On the board...
n = 14 terms.
"Yeah! We got it!" Scootaloo says.
"Hey, I have something to say... It is strange that Cheerilee hasn't noticed that box yet..." Sweetie Belle notices and says, but Cheerilee did not hear they have just said.
"Oh yes! The box! I-- Who opened it?" Cheerilee realises and turns to the box, and becomes a little shocked that somepony had opened it. The calculator lies on top of all the polystyrene wrappings. The class all point to the troubled foals while the troubled ones point to the Cutie Mark Crusaders.
"Oh, so it was you two. Tell me what is this contraption," Cheerilee commands them, and Silver Spoon and Diamond Tiara all walk up to the desk where the calculator lays. The two all start pressing buttons, and nothing works.
"I don't know what it is... There is a lot of buttons on there, and they are too small for hooves," complains Silver Spoon.
"Scootaloo might know what it is," Diamond Tiara points at Scootaloo.
"Wait, me? I have never seen this!" Scootaloo objects, but Diamond Tiara places the TI-92 on Scootaloo's desk anyway.
"Now, try and get this thing working," Diamond Tiara says with an authoritarian voice.
Scootaloo scans the calculator with her eyes jumping from corner to corner and side to side.
"There are so many buttons..." Scootaloo starts to be overwhelmed, but she finds the ON button on the bottom left corner and presses it. The screen then displays an empty calculaton window.
"I turned the thing on..." Scootaloo says, and at that point, the whole class crowds around her, staring at the calculator.
"Try pressing a button!" Sweetie Belle yells. Scootaloo moves her hoof, and presses the 5 button, and what comes up on the screen is 7589.
"I tried pressing 5 only, but my hoof's taking up too much space," Scootaloo complains, but Apple Bloom hands her a pencil. Scootaloo grabs the pencil with her mouth and uses it as a small stylus for the buttons. Scootaloo then strikes the CLEAR button, and the calculator does what she expects it to.
"It has the plus, minus, divide and multiply buttons! Try giving it an problem!" Sweetie Bell advises.
"Let's try 55 + 12 and let's see if it does come up with 67," Scootaloo types "55+12" on the calculator and presses enter, and to her surprise, it did return the correct answer of 67.
