Solving Numeric Patterns with Equal First Differences

Recently at school during mathematics we were dealing with Numeric Patterns, specifically ones with equal first differences, basically meaning patterns where if you had x values of 1, 2, 3 etc. their respective y values would increase by the same amount each time, for example they could be 2, 4, 6 etc. The equation given to my class when finding a formula for these was Tn = a+(n-1)d where a is the y value of 1 and d is the common difference. 

However I thought that there simply must be a simpler way of solving the sum so I went back to basics; what would cause the common difference. I noticed that the formula always involved multiplying or dividing by this difference, and this is because when you times two consecutive numbers by the same thing the second one is always bigger by the amount you multiplied by. 

So I decided you have to times by the common difference if there was a bigger y value than the original x value, and divide by this difference if the y value was smaller. However this usually doesn't get you to the final y value, so you have to do something else. Then I thought about it some more and came up with the idea that the only logical conclusion would be to simply look at the y value and plus or minus whatever you needed to to get there. And this works, leaving a simpler equation of y = x x/÷ d +/- w with everything meaning the same as with the other sum except for w simply meaning whatever works.

Basically, if there is a difference of a, times or divide the input (x) by a and see what you need to add to make the output (y), calling it b. To show you that I'm not just talking bull, let's look at an example;

Input (x):      1   2   3   4   5   6   7   102

Output (y):   5   9  13 17 21 25 29  409

Let's work out the formula using my method, starting with the x value of 1. y = x x/÷ d +/- w. This would then become the working formula; y = x x 4 + 1. You can try it for yourselves!



While this is not something I usually talk about here on my blog, I hope it helps someone else in the future. I would just like to point out that another boy in my class also pointed it out to the teacher, and as he had not seen my book as far as I am aware of I think it is fair to assume that he also figured this out, which implies that many others could also have figured it out. However, for those who haven't I hope this helps you. Please assist me, my blog, and your friends by sharing this post with them if you think it could be helpful. As always, please comment your thoughts and ideas below and I will attempt to answer you. Thanks for reading! Noah  '''⌐(ಠ۾ಠ)¬'''

Comments

  1. JasperThePersonMay 18, 2021 at 4:43 AM

    I also figured this out, I was surprised when we got to the end of the module without being taught this, I thought they would make sure we knew the longer way and understood it, then tell us this method which I had been using the whole time. This happened twice as I was taught it in 2 different schools, they really should show this to us or explain it better, because the first time I tried to show it my teacher didn't seem to understand or want to try it.

    ReplyDelete
    Replies
    1. Noah MTMay 29, 2021 at 11:24 PM

      My teacher was (thankfully) quite happy with us using this method as long as we also knew the other.

      Delete

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