Some Series Counting

 
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 started 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:

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