In an arithmetic progression (AP), the $n$-th term is given by:

$T_n = a + (n-1)d$

where $a$ is the first term and $d$ is the common difference.

Step 1: Use the condition on the 8th and 3rd terms

We are told that the 8th term is three times the 3rd term. Using the formula for the $n$-th term:

$T_8 = a + 7d$ $T_3 = a + 2d$

The condition given is:

$T_8 = 3 \times T_3$

Substitute the expressions for $T_8$ and $T_3$:

$a + 7d = 3(a + 2d)$

Expanding the right-hand side:

$a + 7d = 3a + 6d$

Step 2: Solve for $a$ and $d$

Rearrange the equation to bring terms involving $a$ and $d$ to one side:

$a + 7d - 3a - 6d = 0$ $-2a + d = 0$

Thus, we have:

$d = 2a$

Step 3: Express the sum of the first 8 terms

The sum of the first $n$ terms of an AP is given by the formula:

$S_n = \frac{n}{2} \times (2a + (n-1)d)$

For the sum of the first 8 terms ($S_8$):

$S_8 = \frac{8}{2} \times (2a + 7d)$ $S_8 = 4 \times (2a + 7d)$

Substitute $d = 2a$:

$S_8 = 4 \times (2a + 7 \times 2a)$ $S_8 = 4 \times (2a + 14a)$ $S_8 = 4 \times 16a = 64a$

Step 4: Express the sum of the first 4 terms

Similarly, for the sum of the first 4 terms ($S_4$):

$S_4 = \frac{4}{2} \times (2a + 3d)$ $S_4 = 2 \times (2a + 3d)$

Substitute $d = 2a$:

$S_4 = 2 \times (2a + 3 \times 2a)$ $S_4 = 2 \times (2a + 6a)$ $S_4 = 2 \times 8a = 16a$

Step 5: Show the required condition

We are asked to show that the sum of the first 8 terms is four times the sum of the first 4 terms, i.e.,

$S_8 = 4 \times S_4$

Substitute the values of $S_8$ and $S_4$:

$64a = 4 \times 16a$

This is true, confirming the given condition.

Conclusion

Thus, the sum of the first 8 terms is indeed four times the sum of the first 4 terms, as required.

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Would you like more details on any part of this solution or have any questions?

Here are five related questions to extend your understanding:

1. How would the solution change if the 8th term was twice the 3rd term?
2. How do you derive the formula for the sum of the first $n$ terms of an arithmetic progression?
3. If the sum of the first 6 terms of the AP is 60, how can you find $a$ and $d$?
4. What are some other real-life applications of arithmetic progressions?
5. How does the common difference $d$ affect the overall progression of an AP?

Tip: Always double-check arithmetic progression conditions (e.g., given term relationships) to simplify the equation solving!