I Think

AP

Inspired by the likes of khanacademy and a similar initiative at work, me and my friend Joe were discussing about how to make high school math intuitive and interesting. I was telling him that one of the ways would be to make explanations visual, giving the (x+y) squared example shown below.

Anyway, I had this idea about APs that I told him, and thought was neat enough to share. From this point I am going to assume you readers are a bunch of high schoolers, that is you have an intellectual capacity amounting to that of Jar Jar Binks. So stop playing. You there, thats right you, remove that pen from your nose. And pay attention, or you’ll be in detention.

Do you remember how to find the sum of first n numbers? I am sure you do, and no problem if you dont, we will get there.

Lets say you want to find the sum of first 5 numbers. I am going to show each number by a corresponding number of X’s, like this.

To get the sum of the numbers 1 to 5 you could just count the total number of X’s here. Or you could do what I do here.

Just for the fun of it, I am going to make a copy of the X’s here, I can see you pointing out that I now have twice the number of X’s and am simply getting nowhere. Patience padawan.

How about arrange the two copies side by side, you know like this.

What have we got here? A nice little rectangle, with 5 rows and each row having 6 X’s, giving a total of, that’s right, 5 times 6 X’s, 30, which you pointed out earlier is twice the number of X’s we had in the first place.
Now you could do this for finding the sum of first 6, 7, 8 numbers and find that the sum is half of (6 times 7), (7 times 8 ) and (8 times 9), and for any n it would be half of n times (n+1).

Now natural numbers is just another AP. Let us try and figure out the sum of an AP, say 2, 5, 8, 11.

I am going to do the same thing I did before, represent numbers using X’s, create a copy and form a rectangle.

Observations,
1. we have a row for each item in the series, four rows here
2. each row has the same number of X’s.
3. the total number of X’s in the rectangle is 4(the number of terms in the series) times the number of X’s in one row.
4. curiously, the number of X’s in the first row is the sum of the first term and the last term, ie 2 + 11(the number of X’s in the second row is the sum of second term and penultimate row and so on, but for now let us be happy with the first row)
5. total X’s = 4 times (2 + 11) or in general terms number of terms in the series * (first term + last term), or if you like number of terms * (second term + penultimate term) etc.. -> which is twice the sum we needed in the first place(remember we made a copy before).

Now, I cant think of any real world applications of AP, I have never had to find the sum of a series. Bugs me why I learnt this in the first place. Damn, thats another useless thing, reminds me of trigonometry. Anyway, Joe seems to have caught up this idea of doing videos on math so check his blog if you are interested(dont think he has done one yet). As for me, i will stick to posting mundane stuff, did i tell you about the mole I have in my back and dint know all this time…

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