The question asks to find the first term in an arithmetic progression (AP) that has 4 digits. The given sequence is:

$8, 11, 14, \dots$

In this arithmetic progression:

• The first term $a = 8$
• The common difference $d = 11 - 8 = 3$

The formula for the $n$-th term of an arithmetic progression is:

$a_n = a + (n - 1) \cdot d$

We are asked to find the value of $n$ such that $a_n$ is the first term with 4 digits. This means $a_n \geq 1000$.

Substitute the values for $a$ and $d$:

$a_n = 8 + (n - 1) \cdot 3$

We want $a_n \geq 1000$:

$8 + (n - 1) \cdot 3 \geq 1000$

Now, let's solve for $n$.The equation gives $n = \frac{995}{3}$, which is approximately 331.67. Since $n$ must be an integer, we round up to the next whole number, which is $n = 332$.

Thus, the first term of the arithmetic progression with 4 digits occurs when $n = 332$.

Do you want further clarification or details on this?

Here are 5 related questions to explore:

1. How can I verify that $n = 332$ is the correct value?
2. What is the general formula for the sum of an arithmetic progression?
3. How does changing the common difference $d$ affect the $n$-th term?
4. How can I find the number of terms in an arithmetic progression?
5. What if the progression had a negative common difference?

Tip: Always remember to check whether the result for $n$ needs to be rounded up or down based on the problem requirements.