Let's break down and solve each part of the problem based on the provided image.

Given:

A geometric series is provided:

$\frac{1}{1 - x} + \frac{1}{1 - \frac{3}{2}x} + \frac{1}{1 - \frac{7}{4}x} + \cdots$ with $x > 0$. Now let's go step by step:

a. Find the value of $x$:

This part asks for the value of $x$ where the series converges or follows a pattern. Given the nature of the series, this looks like a geometric progression with variable ratios. I'll analyze the series and identify any constraints on $x$ from the first few terms.

b. Find the value of $3x^2 - 7x$:

After finding $x$ from part (a), we substitute it into the expression $3x^2 - 7x$ and calculate the result.

c. Find $2xr^2$ where $r$ is the ratio of the series:

The problem states that $r$ is the ratio of the series. Once we determine the common ratio $r$ from the geometric series, we calculate $2xr^2$.

d. Find the sixth term of the series:

In a geometric series, the general term for the $n$-th term is:

$T_n = a \cdot r^{n-1}$

where $a$ is the first term and $r$ is the common ratio. We need to use this formula to find the sixth term.

e. Find the seventh term of a new geometric series:

In this part, we are asked to consider a new geometric series where the first term is the same as the original series and the common ratio is $x$. We calculate the seventh term using the formula for the general term of a geometric series:

$T_7 = a \cdot x^6$

I will now begin solving these parts. Let's start by finding the value of $x$ and move step-by-step through the questions.For part (b), the expression $3x^2 - 7x$ simplifies to:

$x(3x - 7)$

This is the simplified form, and the specific value can be determined once we know the value of $x$.

Now, I will proceed with solving part (a) by analyzing the geometric series and finding the value of $x$.For part (a), the series provided is complex, but it follows a geometric progression with variable terms. The pattern suggests that the general term could depend on both $x$ and some ratio involving fractions. Solving this fully requires analyzing convergence conditions or explicit patterns in the series.

Let me work through an analysis to better understand the first few terms and develop a solution. Could you clarify if you expect a general formula for $x$, or should we focus on identifying terms and simplifying specific parts?