Sylph Angel

Sunday, September 19, 2010

Today, I spent the whole afternoon (after tuition) trying to figure out a straightforward method to find the equations of number patterns. You see, I was getting sick of the luck-based guessing and checking method

and the tedious simultaneous equations, so I tried to figure out something new. I thought about linear and quadratic number pattern formulas. You see, the difficulty is not with finding the next few numbers, it's with finding the "master key" of the pattern.

Linear Patterns (an+b) a and b are constants, n is the term number.
Telltale sign: Difference between consecutive numbers is the same. (5,7,9,11,13 - difference is +2)

Pattern: 5, 7, 9, 11, 13...
Step 1. Find the increase/decrease between each consecutive number. (7-5=+2) This is the "a" value.
Step 2. Substitute "a" into the equation. (2n+b)
Step 3. Let n be the term number, and equate the equation to the corresponding term value, to get the value of "b". (2(1)+b=5, b=3)
Equation --> 2x+3
Try it out! It works.

Quadratic Patterns (an2+bn+c) a, b and c are constants, n is the term number.
Telltale sign 1: Difference between consecutive terms increases for positive numbers (2, 5, 10, 17, 26) and decreases for negative numbers (-2, -5, -10, -17, -26).
Telltale sign 2: Difference between consecutive terms is linear (refer to above).

Pattern: 2, 5, 10, 17, 26...
Step 1. List out the differences between the terms in increasing order. (5-2, 10-5, 17-10, 26-17...) --> (3, 5, 7, 9)
Step 2: Find the equation of the linear increase, with steps above. (2n+1)
Step 3: INTEGRATE the equation of the linear increase. (I(2n+1)dn=n2+n+c)
Step 4: Look at the "n2+n+c"
Step 5: Ignore "c" for now.
Step 6: "n2" is the first part of your equation (a=1).
Step 7: Subtract "an" from n. (n-1n=0n) "0n" is the second part of your equation.
Step 8: You now end up with 1n2+0n+c.
Step 9: Substitute in an "n" value and equate the equation to the corresponding term, to find "c". (1(1)2+0(1)+c=2, c=1)
Equation --> n2+1
Cool, huh?

Now I'll attempt something a little more complicated.

Pattern: 1, -4, -12, -23, -37...
Difference: -5, -8, -11, -14
Difference equation: -3n-2
Difference integrate: -1.5n2-2n+c
Value of "a": -1.5
Value of "b": -2-(-1.5)=-0.5
Value of "c": -1.5(1)2-0.5(1)+c=1, c=3
Equation --> -1.5n2-0.5n+3

LOVELY!!!!

Oops, it's a little difficult to understand because of the lack of superscript.

Note: This isn't a proper method. The proper way is to multiply the equation of the linear increase by x, then divide the x2 value by two, and subracting the x2 value from the x value.

vεﯠαחƒőχ @ 9:43:00 PM.