Sunday, October 25, 2009

Some Series Counting

Some Series Counting - Sheva Apelbaum 
At school, we recently started doing pattern progression and regression problems (my dad calls it an arithmetic series).  These problems typically look like this:

87, 67, 47, 27

For some of them, I could find the missing increment by guessing the value, which is especially easy for the obvious ones (Its 20 for the example above).

With longer sequences, it’s not as easy.  So I tried to come up with a formula that I could use to find a solution to any pattern series, no matter  how complicated. I stared by writing out this simple sequence:

5,10,__, 20

I tried several methods (like averaging the sum or adding the sum of the numbers and then dividing it by the total number of spots in the series),  but they didn’t work. 

I then tried subtracting the last value from the first one  in the series (so: 20-5) and noticed that the result (15) was divisible by the number of spaces (3) before the last number.   So I divided 15/3  and got 5, which gave me the increment value.  I tried this formula on 2 other sequences and sure enough, it worked perfectly. 

So here it is,  Sheva’s formula for finding the increment value in a pattern series that uses either addition or subtraction:

Sheva's arithmetic Series forumla

Where:
IV =  increment value
Nl =  the value of the last number in the sequence
Nf=   the value of the first number in the sequence
Ns=  the number of sequences before the last value 

Sample Problem:
Find the missing increment value for the following series
12, __, __, 48, __, __, __96, __ 

Solution:
Nl =    96
Nf =    12
Ns=      7

Increment value = (96 – 12) = 84
Increment value = 84/7
Increment value = 12

Plugging 12 and solving gives us:
12, 24, 36, 48, 60, 72, 84, 96

I knew that you’d want to try some too (you do, don’t you?), so here are a few plus a more challenging one with a beautiful solution at the end. Have fun!

26, __, __, __, __, 91
62, __, __, __, __, __, 104
__, 91, __, __, __, -13, __, __, __, -117

6 comments:

  1. Nice photograph, I'm curious, how did you do that?

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  2. Dear lovely Sheva, I stumbled on your blog some days ago and I must say I am completly hooked and astonished! You are such a special young girl - it is unbelievable! How did you get so smart, so curious, so eager to learn, to try, to do, to figure out? And at the same time being such a friendly, sweet girl with this wonderful smile which one just cannot resist?
    You have lost me completly with all this progression and algebra and... (but I can make nice bags though...)
    The only thing I can m a y b e match you is with the puzzle - maybe - (oh, but surely NOT 5000 pieces, oh no!)
    Ah yes and once I paricipated in a competition where you had to estimate how many bubble gum balls have been in a big yar and I guessed almost bulls eye (I had some little system) and got the prize (don't remeber what that was - do you have a formular to help with memory loss?).
    I will hop in your blog from now on regularly, that's something to look forward to! I hope you allow me to write you from time to time and if you would answer me once in a while I would be honored.
    Now I want to ask you for some help, could you please tell me what that means: "What's cooking in the neck of the woods"? Someone wrote this to me and I have no idea - if you have neither, you could ask your mom...
    For now big hug and kol ha kavod to you! Yael.

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  3. Okay metuka, I figured out two:
    26-39-52-65-78-91
    62-69-76-83-90-97-104
    but could you please help me with the last one?
    Nishika, Yael.

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  4. Yes! I have got the last one too - but with some help I have to admit, from my husband (no wonder HE studied mathematics at the university!):
    117 91 65 39 13 -13 -39 -65 -91 -117
    Very nice!
    Yael.

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  5. Hi Yael,

    Thank you for the absolutly AMAZING comments on my blog! You get 3 out of 3 on your math quiz. I'm very impressed, even if your husband helped you a little.

    I'm so psyched that you're enjoying my blog! Thank you for the compliments.

    To answer your question, "what's cooking in your neck of the woods" means "what's happening on your side of the world". I was filled with curiosity, so I just checked out your Etsy shop and I love it! Especially your backpack with butterflies!

    Todah
    --Sheva :-)

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  6. Thank you so much Sheva - I am looking forward to your next entry!
    Yael.

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