the first and the last term of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression


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!

The first and the last term of an arithmetic progression are 12 and 48 respectively. The sum of the first four terms is 57. Find the number of terms in the progression.

Solve this arithmetic progression problem where the first term is 12, the last term is 48, and the sum of the first four terms is 57. Learn how to apply the sum formula for arithmetic sequences and find the number of terms. This solution is suitable for students in Grades 9-11.

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