Question

Define the word conjecture with an example.

Answer 1

Hint:First, we will define what conjecture is. Along with it, we will write a technical definition and also in simple terms. After that we will write an example, we will use or assume a recurring pattern we will write it for at least three variables and eventually generalize it.

Complete step by step answer:
So, in the question, we are asked to define the word conjecture. So, conjecture is a proposition which is consistent with known data but has neither been verified nor shown to be false. It is synonymous with the hypothesis. In simple terms, a statement that might be true, which is based on some research or reasoning, but is not proven.
We said that it is synonymous with hypothesis but it does not mean that they are similar conjecture is not stated in as formal, or testable, way. So, conjecture is like an educated guess, it is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds true for many cases does not mean that the pattern will hold true for all cases. Conjectures must be proved for mathematical observation to be fully accepted.
When a conjecture is rigorously proved, it becomes a theorem. A conjecture is an important step in problem-solving; it is not just a tool for professional mathematicians. In everyday problem solving, it is very rare that a problem's solution is immediately apparent. Instead, the problem-solving process involves analyzing the problem structure, examining cases, developing a conjecture about the solution, and then confirming that conjecture through proof.
Now, conjectures can be made by anyone, as long as one notices a consistent pattern.
Let’s take an example:
Let’s observe the following pattern:

Now, ${{x}_{n}}$ be the number of segments that connect an \[n\times n\] square lattice. Conjecture an expression for ${{x}_{n}}$. Now, let’s observe the cases given by counting how many segments are in each figure so we have: ${{x}_{0}}=0,{{x}_{1}}=4,{{x}_{2}}=12$, we see that from these three cases, no obvious pattern emerges. Observe what the next case looks like for $n=3$,

Let’s count the segments we have: ${{x}_{3}}=24$ , one might notice that each difference between consecutive terms in the sequence is a multiple of $4$ :
$\begin{align}
  & {{x}_{1}}-{{x}_{0}}=4 \\
 & {{x}_{2}}-{{x}_{1}}=8 \\
 & {{x}_{3}}-{{x}_{2}}=12 \\
\end{align}$
So, this observation could lead one to write a recurrence relation as follows: ${{x}_{n}}={{x}_{n-1}}+4n$ . However, this can become a very tedious calculation if one was required to find the ${{100}^{th}}$ term in the sequence. It would be easier to develop an expression for ${{x}_{n}}$ purely in terms of $n$ .
Now, let’s see that the value of ${{x}_{3}}$ and it can be written as ${{x}_{3}}=24=2\left( 3 \right)\left( 4 \right)$, let ‘s see other cases if they can be written in the same pattern:
$\begin{align}
  & {{x}_{0}}=2\left( 0 \right)\left( 1 \right)=0 \\
 & {{x}_{1}}=2\left( 1 \right)\left( 2 \right)=4 \\
 & {{x}_{2}}=2\left( 2 \right)\left( 3 \right)=12 \\
\end{align}$
We see that pattern appears to hold , so this gives enough information to write a conjecture. So our conjecture will be:
The number of segments connecting an $n\times n$ lattice is defined by the sequence ${{x}_{n}}=2n\left( n+1 \right)$ .

Note:
The last expression can be proven as follows:
One could attempt to observe more cases in the sequence to see if any numerical pattern emerges. So, a better way to tackle these kinds of problems is to think more creatively about how the problem is structured. Let’s observe the same case for $n=3$, except now the horizontal and vertical segments are color-coded.

Notice that there are $4$ horizontal lengths (in red), and each of them consists of $3$ segments. The same is true for vertical lengths (in green). So, now ${{x}_{3}}=2\left( 3 \right)\left( 4 \right)=24$ , now further procedure will be the same.