The problem is about a geometric sequence, where the sum of the first and third terms is 5, and the sum of the fourth and sixth terms is 40. We need to find:

a. The first term and the common ratio
b. The 10th term

Step-by-step solution:

Let the first term be $a$ and the common ratio be $r$.

For a geometric sequence, the $n$-th term is given by: $T_n = a \cdot r^{n-1}$

Given conditions:

1. The sum of the first and third terms is 5: $a + a \cdot r^2 = 5$ This simplifies to: $a(1 + r^2) = 5 \quad \text{(Equation 1)}$

2. The sum of the fourth and sixth terms is 40: $a \cdot r^3 + a \cdot r^5 = 40$ This simplifies to: $a(r^3 + r^5) = 40 \quad \text{(Equation 2)}$

Solving the system of equations:

We now have two equations:

1. $a(1 + r^2) = 5$
2. $a(r^3 + r^5) = 40$

Let's solve this system to find $a$ and $r$.

I'll calculate the solution.The solution to the system of equations gives us the following values for $a$ (the first term) and $r$ (the common ratio):

1. Real solution:
   $a = 1$, $r = 2$

The other solutions involve complex numbers, but since we're dealing with a geometric sequence in real numbers, the valid solution is:

• $a = 1$
• $r = 2$

Part (b): Finding the 10th term

The 10th term of a geometric sequence is given by: $T_{10} = a \cdot r^{9}$ Substituting $a = 1$ and $r = 2$, we can calculate $T_{10}$. Let me compute that now.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate the 10th term by using the formula $T_{10} = a \cdot r^9$, where $a = 1$ and $r = 2$. So:

$T_{10} = 1 \cdot 2^9 = 512$

Would you like more details on any part of the solution? Feel free to ask.

Here are 5 related questions for deeper understanding:

1. What is the general formula for the $n$-th term in a geometric sequence?
2. How do you find the sum of terms in a geometric progression?
3. What is the difference between an arithmetic and geometric sequence?
4. Can you derive the sum of the first $n$ terms of a geometric sequence?
5. How can you determine whether a sequence is geometric just by looking at the terms?

Tip: In geometric sequences, multiplying each term by the common ratio gives the next term, making it easy to spot exponential growth.