Given that the sum of four terms of an arithmetic progression (A.P.) is 24, let's find the extreme, mean, and each term.

Step 1: Understanding the A.P.

An A.P. is a sequence of numbers in which the difference between any two consecutive terms is constant. Let the four terms of the A.P. be $a_1$, $a_2$, $a_3$, and $a_4$. We know that:

$a_1 + a_2 + a_3 + a_4 = 24$

Step 2: Expressing the Terms in Terms of the First Term and Common Difference

The terms of the A.P. can be expressed as: $a_1 = a$ $a_2 = a + d$ $a_3 = a + 2d$ $a_4 = a + 3d$

Here, $a$ is the first term and $d$ is the common difference.

Step 3: Setting Up the Equation

The sum of the four terms is: $a_1 + a_2 + a_3 + a_4 = a + (a + d) + (a + 2d) + (a + 3d) = 4a + 6d$ We are given that: $4a + 6d = 24$

Step 4: Simplifying the Equation

Divide the entire equation by 2: $2a + 3d = 12$

Step 5: Determining the Extreme and Mean Terms

• Extreme Terms: These are the first term $a_1$ and the fourth term $a_4$.
• Mean Terms: These are the second term $a_2$ and the third term $a_3$.

Step 6: Solving for $a$ and $d$

There isn't enough information to uniquely determine $a$ and $d$, but we can express $d$ in terms of $a$: $d = \frac{12 - 2a}{3}$

Step 7: Finding the Terms

Now, let's write the terms in terms of $a$:

• First term: $a_1 = a$
• Second term: $a_2 = a + \frac{12 - 2a}{3}$
• Third term: $a_3 = a + 2 \times \frac{12 - 2a}{3}$
• Fourth term: $a_4 = a + 3 \times \frac{12 - 2a}{3}$

These represent the terms in terms of $a$.

Summary:

• First term (a_1): $a$
• Second term (a_2): $a + d$
• Third term (a_3): $a + 2d$
• Fourth term (a_4): $a + 3d$
• Extreme terms: $a_1$ and $a_4$
• Mean terms: $a_2$ and $a_3$

Would you like more details on any of the steps, or do you have any other questions?

Relative Questions:

1. How can the sum of the first $n$ terms of an A.P. be calculated?
2. What happens to the sum of terms of an A.P. if the common difference $d$ is negative?
3. Can an arithmetic progression have all terms equal? What does it imply about the common difference?
4. How do you determine the common difference if only the first term and the last term are known?
5. In an A.P., if the first term and the sum of terms are known, how do you find the common difference?

Tip:

Remember that in an arithmetic progression, knowing any two pieces of information (like the sum of terms, common difference, or specific terms) can help determine the entire sequence.