# Math: How Do I Find The Pattern's Rule? **URL:** \ **Category:** Factual Questions\ **Created:** [September 10, 2004, 11:49am UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903 "2004-09-10T11:49:37Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [September 10, 2004, 11:49am UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/1 "2004-09-10T11:49:37Z")
I can’t recall how to find the general rule when given the following set of numbers: 0, 3, 10, 21, 36… Or, in a table: for n=1, the result = 0 for n=2, the result = 3 for n=3, the result = 10 for n=4, the result = 21 for n=5, the result = 36 First, I found the first difference between all consecutive values which varies as: for n=1, result = n/a for n=2, result = 3 for n=3, result = 7 for n=4, result = 11 for n=5, result = 15 Then, I found the second difference (delta between the first differences), and we find a constant of k=4 Because it took two steps to get a constant delta, we know the general pattern is an^2 + bn + c…but I can’t solve it! Does anyone recall how to do this? One example in a geometry book suggests to notice that the results follow the following pattern: 1_3, 2_5, 3_7, 4_9… so the pattern is something like n(n^2 + c)…but I still can’t get it to work. Last, what kind of series or progression is this? - Jinx ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 10, 2004, 11:57am UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/2 "2004-09-10T11:57:13Z")
2n[sup]2[/sup] - 3n + 1 ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [September 10, 2004, 12:03pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/3 "2004-09-10T12:03:35Z")
> [@Complex Conjugate](#): > > 2n[sup]2[/sup] - 3n + 1 OK, that’s helpful. Now, can you explain how you found this solution? We’d like to learn a little if it’s not asking too much! - Jinx ---
**Author:** ![Complex\_Conjugate](https://avatars.discourse-cdn.com/v4/letter/c/3bc359/32.png) [@Complex\_Conjugate](https://boards.straightdope.com/u/Complex_Conjugate)\ **Post date:** [September 10, 2004, 12:20pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/4 "2004-09-10T12:20:01Z")
You already have your answer: (0_1), (1_3), (2_5), (3_7), (4\*9), ((n-1)(2n-1)) ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [September 10, 2004, 12:28pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/5 "2004-09-10T12:28:24Z")
> [@Complex Conjugate](#): > > You already have your answer: > (0_1), (1_3), (2_5), (3_7), (4\*9), ((n-1)(2n-1)) I guess…but really it seems like, once you get to this step, you’re mentally asking yourself the difference between “n” and each factor. How would you explain the general technique to a high schooler…lucky guessing until it works? - Jinx ---
**Author:** ![Jinx](https://avatars.discourse-cdn.com/v4/letter/j/c6cbf5/32.png) [@Jinx](https://boards.straightdope.com/u/Jinx)\ **Post date:** [September 10, 2004, 12:30pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/6 "2004-09-10T12:30:30Z")
Oops! The previous heading was typed in haste. Yes, I see it works for all terms in the sequence. Thanks for your help… - Jinx ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [September 10, 2004, 12:52pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/7 "2004-09-10T12:52:35Z")
First off, this sort of sequence would generally be referred to as a quadratic sequence. As far as a general method for solving it, once you know that it follows the pattern s(n) = a(n^2) + bn + c, you can use any three elements of the sequence (such as the first three) to solve for a, b, and c, just like any other time you’re solving a system of three equations in three unknowns. To run through one such solution quickly: for n = 1: (1) a + b + c = 0 for n = 2: (2) 4a + 2b + c = 3 Subtracting (1) from (2): (3) 3a + b = 3 for n = 3: (4) 9a + 3b + c = 10 Subtracting (1) from (4): (5) 8a + 2b = 10 Multiplying (4) by a factor of 2: (6) 6a + 2b = 6 Subtracting (6) from (5): (7) 2a = 4, which is easily solved as a = 2 Substituting the new value of a into (6): (8) 12 + 2b = 6, which is solved as b = -3 Substituting the values of a and b into (1): (9) 2 - 3 + c = 0, which is solved as c = 1 Therefore the general equation is 2(n^2) -3n + 1 Hope that helps. ---
**Author:** ![Giles](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/giles/32/60_2.png) [@Giles](https://boards.straightdope.com/u/Giles)\ **Post date:** [September 10, 2004, 1:03pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/8 "2004-09-10T13:03:16Z")
There’s another explanation of the series, which gives the same results as earlier posters. It’s every other member of the series of triangular numbers: 0, [1], 3, [6], 10, [15], 21, [28], 36, [45], … The differences between members of this series are just the natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, … Note that if you add these in pairs, you get the differences in the original series: 1+2=3, 3+4=7, 5+6=11, 7+8=15, … The usual formula for the triangular numbers is: n(n-1)/2 If you add any two consecutive members of the series, you get a square, e.g. 28+36 = 64 = 8 squared ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [September 10, 2004, 1:47pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/9 "2004-09-10T13:47:30Z")
It looks like the OP’s question is a problem in “the calculus of finite differences.” For a good short introduction, see Chapter 2 of Martin Gardner’s _Colossal Book of Mathematics_. ---
**Author:** ![chrisk](https://avatars.discourse-cdn.com/v4/letter/c/6de8d8/32.png) [@chrisk](https://boards.straightdope.com/u/chrisk)\ **Post date:** [September 10, 2004, 2:22pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/10 "2004-09-10T14:22:15Z")
Minor problem with my solution… the explanation for equation (6) should read: Multiplying ( **3** ) by a factor of 2: Makes a bit more sense that way. 😃 (Do I get any marks taken off for that mistake? :smack: ) ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 10, 2004, 7:45pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/11 "2004-09-10T19:45:14Z")
> [@Jinx](#): > > Because it took two steps to get a constant delta, we know the general pattern is an^2 + bn + c…but I can’t solve it! Plug in n=1 and n=2, and n=3 to get linear equations for a, b, and c > [@](#): > > what kind of series or progression is this? Nothing special. Just the evaluations of a polynomial at the sequence of positive integers. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [September 10, 2004, 7:49pm UTC](https://boards.straightdope.com/t/math-how-do-i-find-the-patterns-rule/263903/12 "2004-09-10T19:49:38Z")
Another solution: [look it up](http://www.research.att.com/~njas/sequences/).