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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.

The sum of the first and third terms of a geometric sequence is 5, and the sum of the fourth and sixth terms is 40. Determine: a. The first term and the common ratio, b. The 10th term.

Solve a geometric sequence problem where the sum of the first and third terms is 5, and the sum of the fourth and sixth terms is 40. Learn how to find the first term, common ratio, and the 10th term with detailed steps.

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