# What is the general stragety for conjecturing a formula based off a pattern?

Is it simply to guess and evolve a answer until it gets closer or is there an approach?

Ex: Find the formula for:

$a_k = \frac{a_{k-1}}{2} + 1$ where $a_0 = 1$.

One would go: $a_1 = 3/2, a_2 = 7/4, a_3 = 15/8, \dots$ and notice a pattern in the denominator.

However finding the real answer to be $a_n = 2 - \frac{1}{2^n}$ would never cross my mind.

Despite how simple it looks I would never guess the formula for it until I read the solution. Without a calculator 2 infront would have never crossed my mind.

• "Conjecturing a formula based off a pattern" is far, far too broad a problem for there to exist any kind of "general strategy". You shouldn't feel too bad, anyway. Real mathematics is seldom done by guessing the answer based on a pattern, and even when you do spot something, it's as much luck as it is ability. Focus on other mathematical skills. – Jack M Feb 27 '14 at 23:46
• @JackM yes there are programs out there that do it. What strategy do they use? – Sad CRUD Developer Feb 27 '14 at 23:47
• Usually they look for a very specific type of formula, so specific that the exact formula can be solved algebraically. But it won't always be the "right" formula (for example, it'll rarely be the simplest one). – Jack M Feb 27 '14 at 23:49
• That will be an interesting question. How machines can conjure formulas. I saw wolfram in action and it amazes me. There HAS to be a algebraic stagety – Sad CRUD Developer Feb 27 '14 at 23:51
• @JackM: I disagree that real mathematics is seldom done by guessing the answer based on a pattern. It obviously depends on what field you are in, but many people's research involves trying to piece together complicated patterns and, often, the first proof that is available is a guess-and-check approach. Of course, seeking out the underlying reason behind a pattern is important too. And experienced researchers don't guess blindly; they make educated guesses from an intuition formed from studying the problem for a long time. Practicing the art of recognizing patterns is a very worthwhile task. – Michael Joyce Feb 27 '14 at 23:55

## 2 Answers

The answer is to gain experience by studying patterns in other problems. You learn to recognize patterns through practice. It is okay if you don't recognize how to find the pattern in this example at first, but now that you have been shown the pattern, study it carefully and try to see if you can understand why the pattern emerged and how, in hindsight, you could have noticed it. If you don't go back and understand how this pattern works after it has been revealed to you, you'll be in no better position to discover a pattern the next time you need to.

For example, in this problem, notice that the recurrence relation involves dividing by $2$. Is it any surprise, then, that in the pattern for the sequence $a_n$, the denominator in each successive term is the next power of $2$? The numerator is more subtle, but if you study it carefully, you will uncover its pattern and, much more importantly, the explanation for the pattern as well.

Math is a challenging subject that requires you to always be thinking critically and asking yourself why things are working out the way they do. If you adopt such a mindset for solving problems, you'll start to notice more and more as time goes on.

Well the only remaining thing to notice is that the nominator is one less than twice the denominator.

Then you can write $$\frac{2\cdot2^n-1}{2^n} = 2 - \frac{1}{2^n}$$ and prove it by induction.