	The sum of the first nth terms of an arithmetic series is n/2 (3n-7). If the third term and the seventh term of arithmetic series are the third term and the fourth term of a geometric series respectively, find the sum of the first eight terms of this geometric series.

Let's break down the problem step by step.

### Step 1: Determine the general form of the arithmetic series
We are given the sum of the first \( n \) terms of an arithmetic series as:

\[
S_n = \frac{n}{2}(3n - 7)
\]

The sum of the first \( n \) terms of an arithmetic series is also given by:

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

where:
- \( a \) is the first term,
- \( d \) is the common difference.

### Step 2: Equating and finding \( a \) and \( d \)
Comparing the given sum formula with the standard sum formula, we get:

\[
\frac{n}{2} (2a + (n-1)d) = \frac{n}{2}(3n - 7)
\]

Thus, equating the expressions inside the parentheses:

\[
2a + (n-1)d = 3n - 7
\]

This is a linear equation in terms of \( n \). To determine \( a \) and \( d \), we can find the first term and the common difference by substituting specific values of \( n \).

#### For \( n = 1 \):
\[
2a + (1-1)d = 3(1) - 7
\]
\[
2a = -4 \quad \Rightarrow \quad a = -2
\]

#### For \( n = 2 \):
\[
2a + (2-1)d = 3(2) - 7
\]
\[
2(-2) + d = 6 - 7
\]
\[
-4 + d = -1 \quad \Rightarrow \quad d = 3
\]

So, the first term \( a = -2 \) and the common difference \( d = 3 \).

### Step 3: Calculate the third and seventh terms of the arithmetic series
The general formula for the \( n \)-th term of an arithmetic series is:

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

So:

\[
\text{Third term } (a_3) = a + 2d = -2 + 2(3) = 4
\]
\[
\text{Seventh term } (a_7) = a + 6d = -2 + 6(3) = 16
\]

### Step 4: Set up the geometric series
We are told that the third term of the arithmetic series (which is 4) is the third term of a geometric series, and the seventh term of the arithmetic series (which is 16) is the fourth term of the same geometric series.

Let the first term of the geometric series be \( G_1 = g \) and the common ratio be \( r \).

The third term of the geometric series is:
\[
G_3 = g r^2 = 4
\]

The fourth term of the geometric series is:
\[
G_4 = g r^3 = 16
\]

### Step 5: Solve for \( g \) and \( r \)
We have two equations:
\[
g r^2 = 4 \quad \text{(Equation 1)}
\]
\[
g r^3 = 16 \quad \text{(Equation 2)}
\]

Divide Equation 2 by Equation 1 to eliminate \( g \):
\[
\frac{g r^3}{g r^2} = \frac{16}{4}
\]
Simplifying:
\[
r = 4
\]

Now substitute \( r = 4 \) into Equation 1:
\[
g \times 4^2 = 4 \quad \Rightarrow \quad 16g = 4 \quad \Rightarrow \quad g = \frac{1}{4}
\]

### Step 6: Calculate the sum of the first 8 terms of the geometric series
The sum of the first \( n \) terms of a geometric series is given by:
\[
S_n = g \frac{r^n - 1}{r - 1}
\]
For \( n = 8 \):
\[
S_8 = \frac{1}{4} \times \frac{4^8 - 1}{4 - 1}
\]
Calculate \( 4^8 \):
\[
4^8 = 65536
\]
So:
\[
S_8 = \frac{1}{4} \times \frac{65536 - 1}{3} = \frac{1}{4} \times \frac{65535}{3} = \frac{65535}{12} = 5461.25
\]

### Final Answer
The sum of the first eight terms of the geometric series is \( 5461.25 \).

Would you like any details or have any questions? Here are some related questions you might find interesting:

1. What is the relationship between arithmetic and geometric sequences?
2. How does the common ratio in a geometric series affect the growth rate of the series?
3. How can we find the sum of an infinite geometric series?
4. What happens if the common ratio \( r \) is between -1 and 1?
5. How can we determine if a series is converging or diverging?
6. How can you prove that the \( n \)-th term formula for an arithmetic sequence is linear?
7. What other methods could be used to solve this type of problem?
8. How does changing the initial term affect the sum of the series?

**Tip:** When dealing with series problems, always make sure to clearly define and isolate the variables that are critical to the problem. This will help in managing complex algebraic manipulations more effectively.

Explore how to find the sum of the first eight terms of a geometric series derived from an arithmetic series problem. Understand the connection between arithmetic and geometric sequences with detailed step-by-step calculations and formulas.

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