Question 8

Explanation

Identifying Production Patterns

This problem applies the concept of an Arithmetic Sequence, where pan production increases by a constant amount each month. We have information about production in the $3\text{rd}$ month and the $6\text{th}$ month.

Given:

Determining Production Increase and Initial Production

In an arithmetic sequence, the $n\text{-th}$ term is given by $U_n = a + (n-1)b$. We can form two linear equations to find the initial value $a$ and the common difference $b$.

The difference between these two equations gives the value of $b$:

So, production increases by $250$ units each month. Next, substitute the value of $b$ into equation $(1)$ to find $a$:

Production in the first month is $1,000$ units.

Determining the Target Month

We need to find the $n\text{-th}$ month where production exceeds $10,000$ units.

Simplify the inequality:

Since $n$ must be an integer greater than $37$, the first month that satisfies the condition is the $38\text{th}$ month.

Note that in the $37\text{th}$ month, production exactly reaches $10,000$ units, so it has not yet exceeded the target.