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The problem asks us to find the positive value of \( a \) such that the following equation is satisfied:

\[
\sum_{n=2}^{\infty} 7 a^n = 10
\]

This is an infinite geometric series with the general form:

\[
\sum_{n=2}^{\infty} r a^n = r \sum_{n=2}^{\infty} a^n
\]

We can rewrite the series as:

\[
7 a^2 + 7 a^3 + 7 a^4 + \dots = 10
\]

Factoring out 7, we get:

\[
7 (a^2 + a^3 + a^4 + \dots) = 10
\]

The expression inside the parentheses, \( a^2 + a^3 + a^4 + \dots \), is itself an infinite geometric series with the first term \( a^2 \) and common ratio \( a \). The sum of an infinite geometric series with first term \( A \) and common ratio \( r \) (where \( |r| < 1 \)) is given by:

\[
\frac{A}{1 - r}
\]

In this case, \( A = a^2 \) and \( r = a \), so we can write:

\[
a^2 + a^3 + a^4 + \dots = \frac{a^2}{1 - a}
\]

Substituting back, we have:

\[
7 \cdot \frac{a^2}{1 - a} = 10
\]

Now, solving for \( a \):

1. Divide both sides by 7:
   \[
   \frac{a^2}{1 - a} = \frac{10}{7}
   \]

2. Cross-multiply to eliminate the fraction:
   \[
   a^2 = \frac{10}{7} (1 - a)
   \]

3. Distribute \( \frac{10}{7} \):
   \[
   a^2 = \frac{10}{7} - \frac{10}{7}a
   \]

4. Rearrange the terms to form a standard quadratic equation:
   \[
   a^2 + \frac{10}{7}a - \frac{10}{7} = 0
   \]

5. Multiply through by 7 to clear the fractions:
   \[
   7a^2 + 10a - 10 = 0
   \]

Now, we can solve this quadratic equation using the quadratic formula:

\[
a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 7 \), \( b = 10 \), and \( c = -10 \).

Plugging in these values:

\[
a = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 7 \cdot (-10)}}{2 \cdot 7}
\]

\[
a = \frac{-10 \pm \sqrt{100 + 280}}{14}
\]

\[
a = \frac{-10 \pm \sqrt{380}}{14}
\]

\[
a = \frac{-10 \pm 2\sqrt{95}}{14}
\]

Simplifying further, we get:

\[
a = \frac{-5 \pm \sqrt{95}}{7}
\]

Since we are asked for the positive value of \( a \), we take the positive root:

\[
a = \frac{-5 + \sqrt{95}}{7}
\]

Let me know if you would like further clarification on any steps or a numeric approximation for this solution.

---

Here are 5 related questions for further exploration:

1. How do you find the sum of an infinite geometric series starting from a general term?
2. What is the convergence condition for an infinite geometric series?
3. Can the quadratic formula be used to solve all types of quadratic equations?
4. How does changing the common ratio affect the sum of an infinite series?
5. Why is only the positive root considered in this context?

**Tip:** When working with infinite series, always verify if the common ratio's absolute value is less than 1 to ensure convergence.

Considere la siguiente suma: \( \sum_{n=2}^{\infty} 7 a^n = 10 \). El valor positivo de \( a \) para que se cumpla la ecuación es:

Discover how to find the positive value of \( a \) that satisfies the infinite geometric series equation \( \sum_{n=2}^{\infty} 7 a^n = 10 \). This solution involves key algebraic concepts, including infinite series and quadratic equations, making it ideal for Grades 10-12.

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