To find the number of terms in the arithmetic progression, we are given:

• First term ($a_1$) = 12
• Last term ($a_n$) = 48
• Sum of the first four terms = 57

We know the sum of the first $n$ terms of an arithmetic progression is given by the formula:

$S_n = \frac{n}{2} \left( 2a_1 + (n-1)d \right)$

Where:

• $S_n$ is the sum of the first $n$ terms.
• $a_1$ is the first term.
• $d$ is the common difference.
• $n$ is the number of terms.

Step 1: Find the common difference ($d$)

The sum of the first four terms is 57. So, applying the sum formula for the first four terms:

$S_4 = \frac{4}{2} \left( 2a_1 + 3d \right)$

Substituting the known values:

$57 = 2 \left( 2(12) + 3d \right)$

Simplifying:

$57 = 2 \left( 24 + 3d \right)$

$57 = 48 + 6d$

$6d = 57 - 48 = 9$

$d = \frac{9}{6} = 1.5$

So, the common difference is $d = 1.5$.

Step 2: Find the number of terms ($n$)

Now, we use the formula for the $n$-th term of an arithmetic progression:

$a_n = a_1 + (n-1) d$

We are given that the last term $a_n = 48$, so:

$48 = 12 + (n-1) \cdot 1.5$

Simplifying:

$48 - 12 = (n-1) \cdot 1.5$

$36 = (n-1) \cdot 1.5$

$n-1 = \frac{36}{1.5} = 24$

$n = 24 + 1 = 25$

So, the number of terms in the arithmetic progression is $n = 25$.

Final Answer:

The arithmetic progression has 25 terms.

Would you like further clarification or more details?

Here are five related questions:

1. How is the sum formula for an arithmetic progression derived?
2. Can the sum of an arithmetic progression be negative?
3. What if the common difference is negative? How does that affect the progression?
4. Can the first term of an arithmetic progression be zero? What happens then?
5. How do we solve problems involving geometric progressions?

Tip: Always ensure to substitute values correctly into known formulas to avoid calculation errors!