To find the sum of the first 25 terms of the arithmetic progression, let's start by determining the common difference and the first term of the sequence.

Step 1: Identify the terms

Given:

• The third term $a_3 = 4$
• The fourth term $a_4 = 7$

Step 2: Use the formula for the $n$th term of an arithmetic progression

The general formula for the $n$th term of an arithmetic progression is: $a_n = a_1 + (n-1)d$ where:

• $a_n$ is the $n$th term,
• $a_1$ is the first term, and
• $d$ is the common difference.

For the third term $a_3$: $a_3 = a_1 + 2d = 4$

For the fourth term $a_4$: $a_4 = a_1 + 3d = 7$

Step 3: Find the common difference $d$

Subtract the equation for $a_3$ from the equation for $a_4$: $(a_1 + 3d) - (a_1 + 2d) = 7 - 4$ $d = 3$

Step 4: Find the first term $a_1$

Using $a_3 = a_1 + 2d = 4$: $a_1 + 2(3) = 4$ $a_1 + 6 = 4$ $a_1 = -2$

Step 5: Find the sum of the first 25 terms

The sum of the first $n$ terms of an arithmetic progression is given by: $S_n = \frac{n}{2} \times \left(2a_1 + (n-1)d\right)$ For $n = 25$, $a_1 = -2$, and $d = 3$: $S_{25} = \frac{25}{2} \times \left[2(-2) + (25-1) \times 3\right]$ $S_{25} = \frac{25}{2} \times \left[-4 + 72\right]$ $S_{25} = \frac{25}{2} \times 68$ $S_{25} = 25 \times 34 = 850$

Final Answer:

The sum of the first 25 terms of the arithmetic progression is 850.

Would you like more details or have any questions?

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Here are some related questions you might be interested in:

1. How do you derive the formula for the sum of an arithmetic progression?
2. What happens to the sum if the common difference is negative?
3. How do you find the sum if the first term is unknown?
4. Can you calculate the sum of the first 100 terms of this sequence?
5. What is the general formula for the sum of an infinite arithmetic progression?
6. How do you find the common difference if only the first and last terms are given?
7. What if the sum of the first $n$ terms is given instead of the common difference?
8. How can you determine the number of terms if the sum and the first term are given?

Tip: Always double-check your calculations, especially when working with sequences, to avoid errors that can compound in the final result.