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.