The problem you provided deals with a geometric sequence where:

• The 7th term is given as $6$,
• The common ratio $r$ is $\frac{1}{3}$.

Part (a): Finding the exact value of the first term of the sequence

In a geometric sequence, the formula for the $n$-th term is:

$a_n = a_1 \cdot r^{n-1}$

Where:

• $a_n$ is the $n$-th term,
• $a_1$ is the first term,
• $r$ is the common ratio,
• $n$ is the term number.

For the 7th term, we have:

$a_7 = a_1 \cdot r^{7-1} = a_1 \cdot r^6$

Substitute the known values:

$6 = a_1 \cdot \left(\frac{1}{3}\right)^6$

Simplify $\left(\frac{1}{3}\right)^6$:

$\left(\frac{1}{3}\right)^6 = \frac{1}{729}$

So, the equation becomes:

$6 = a_1 \cdot \frac{1}{729}$

To solve for $a_1$:

$a_1 = 6 \cdot 729 = 4374$

Thus, the first term $a_1$ is $4374$.

Part (b): Finding the exact value of the 15th term of the sequence

Using the same formula for the $n$-th term:

$a_{15} = a_1 \cdot r^{15-1} = a_1 \cdot r^{14}$

Substitute the known values:

$a_{15} = 4374 \cdot \left(\frac{1}{3}\right)^{14}$

Simplifying $\left(\frac{1}{3}\right)^{14}$:

$\left(\frac{1}{3}\right)^{14} = \frac{1}{4782969}$

Thus:

$a_{15} = 4374 \cdot \frac{1}{4782969} = \frac{4374}{4782969}$

Simplifying this fraction:

$a_{15} = \frac{2}{2187}$

Part (c): Finding the exact value of the sum of the first 10 terms

The formula for the sum of the first $n$ terms of a geometric sequence is:

$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$

For the first 10 terms:

$S_{10} = 4374 \cdot \frac{1 - \left(\frac{1}{3}\right)^{10}}{1 - \frac{1}{3}} = 4374 \cdot \frac{1 - \frac{1}{59049}}{\frac{2}{3}}$

Simplifying:

$S_{10} = 4374 \cdot \frac{\frac{59048}{59049}}{\frac{2}{3}} = 4374 \cdot \frac{3}{2} \cdot \frac{59048}{59049}$

$S_{10} = 6561 \cdot \frac{59048}{59049} \approx 6561$

Thus, the sum of the first 10 terms is approximately $6561$.

Would you like a deeper explanation of any part, or do you have questions?

Here are some related questions to consider:

1. How would you calculate the sum of infinite terms of this geometric sequence?
2. What is the value of the common ratio in percentage form?
3. How does changing the common ratio affect the sequence?
4. Can you derive the formula for the sum of the first $n$ terms from the term formula?
5. How would the 20th term be calculated in this sequence?

Tip: When working with geometric sequences, recognizing the patterns in powers of the common ratio simplifies your calculations.