Answer:
(a) The result is 5n.
(b) See below for proof of the conjecture.
Step-by-step explanation:
A conjecture is a statement or proposition that is based on incomplete or insufficient evidence but is considered likely to be true. It is a hypothesis or educated guess that is made in the absence of complete information or proof. Conjectures are often formed through observation, experimentation, or patterns observed in data.
If we observe the pattern of the given data, we can see that when each number goes through the given process, the result is five times the original number. So, the conjecture can be expressed as 5n, where n is the original number.
To prove this, we can use deductive reasoning to show that the result of the given process is consistently 5n, by breaking down the process into individual steps and demonstrating mathematically that the result is five times the original number:
[tex]\begin{aligned}&\textsf{Multiply the number by 10:}&\quad &10n\\\\&\textsf{Add $2$ to the product:}&\quad &10n + 2\\\\&\textsf{Divide this sum by $2$:}&\quad &\dfrac{{10n + 2}}{2}\\\\&\textsf{Subtract $1$ from the quotient:}&\quad &\dfrac{{10n + 2}}{2} - 1\end{aligned}[/tex]
Now, let's simplify the expression:
[tex]\begin{aligned}\dfrac{{10n + 2}}{2} - 1 &= \dfrac{10n}{2}+\dfrac{2}{2}-1\\\\&=5n+1-1\\\\&= 5n\end{aligned}[/tex]
Therefore, the result of the process is 5n, which confirms the conjecture that the result of the process is 5 times the original number.
