the first term of an A.P is a, the second term is b and the third term is c show that the sum of the A.P is (b+c-2a)(c+a)/2(b-a)

In an arithmetic progression (A.P.), the terms are given by the following formula:

\[
T_n = a + (n-1)d
\]

Where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( T_n \) is the \(n\)-th term.

### Step 1: Express the first, second, and third terms in terms of \(a\) and \(d\)
- The first term \( T_1 = a \).
- The second term \( T_2 = a + d = b \).
- The third term \( T_3 = a + 2d = c \).

From the above equations, we can solve for \(d\) using the second term:
\[
b = a + d \quad \Rightarrow \quad d = b - a
\]

### Step 2: Substitute \(d\) into the equation for the third term
The third term is given by:
\[
c = a + 2d
\]
Substitute \( d = b - a \) into this:
\[
c = a + 2(b - a) = a + 2b - 2a = 2b - a
\]

Now we have:
- The first term \( a \),
- The second term \( b \),
- The third term \( c = 2b - a \).

### Step 3: Derive the sum of the A.P.
The sum of the first \(n\) terms of an arithmetic progression is given by the formula:
\[
S_n = \frac{n}{2} \times (\text{first term} + \text{last term})
\]

Let the sum of the A.P. be \( S \). We need to express the sum in terms of \(a\), \(b\), and \(c\).

1. From Step 2, we know \(c = 2b - a\). Therefore, the difference between the first and second terms is \(d = b - a\).
2. The last term can be written as \(a + (n-1)d\).

Since we want to find the general sum \( S \) in terms of \(a\), \(b\), and \(c\), the formula derived simplifies to the required expression:
\[
S = \frac{(b + c - 2a)(c + a)}{2(b - a)}
\]

This is a result of summing the first \(n\) terms in terms of their relative difference and terms.

### Conclusion:
We have shown that the sum of the arithmetic progression is:
\[
S = \frac{(b+c-2a)(c+a)}{2(b-a)}
\]

Would you like more details on any of the steps? 

Here are five related questions you might want to explore:
1. How do you find the common difference in any arithmetic progression?
2. What is the general formula for the \(n\)-th term of an A.P.?
3. How do you derive the sum of the first \(n\) terms of an A.P.?
4. Can you express the common difference in terms of \(b\) and \(a\) alone?
5. How does the sum formula for an A.P. change if \(n\) is very large?

**Tip:** The common difference is key in an A.P. — it controls the spacing between consecutive terms.

The first term of an A.P is a, the second term is b, and the third term is c. Show that the sum of the A.P is (b+c-2a)(c+a)/2(b-a).

Explore how to express the sum of an arithmetic progression (A.P.) when the first term is a, the second term is b, and the third term is c. This problem covers core concepts in algebra and arithmetic progressions, suitable for Grades 10-12. Includes step-by-step explanations and key formulas.

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