https://file.aiquickdraw.com/mathbot/file/17272468553165o5ltxjp.jpg

Let's break down the sequences in the image step by step:

### Problem 1:
The sequence is:  
\[ 5, 10, 20, 40, \dots \]

This is a geometric sequence where each term is multiplied by 2.  
The general formula for the \(n\)-th term of a geometric sequence is:  
\[ U_n = U_1 \times r^{n-1} \]  
Where:
- \(U_1 = 5\) (the first term),
- \(r = 2\) (the common ratio).

#### Finding \(U_8\):
Using the formula:  
\[ U_8 = 5 \times 2^{7} = 5 \times 128 = 640 \]

#### Finding \(S_6\) (Sum of the first 6 terms):
The sum of the first \(n\) terms in a geometric sequence is given by:  
\[ S_n = \frac{U_1 (1 - r^n)}{1 - r} \text{ for } r \neq 1 \]

For \(S_6\):  
\[ S_6 = \frac{5 (1 - 2^6)}{1 - 2} = \frac{5 (1 - 64)}{-1} = 5 \times 63 = 315 \]

### Problem 2:
The sequence is:  
\[ 3, 9, 27, 81, \dots \]

This is also a geometric sequence, where each term is multiplied by 3.  
The general formula is:  
\[ U_n = U_1 \times r^{n-1} \]  
Where:
- \(U_1 = 3\),
- \(r = 3\).

#### Finding \(U_9\):
Using the formula:  
\[ U_9 = 3 \times 3^{8} = 3 \times 6561 = 19683 \]

#### Finding \(S_7\) (Sum of the first 7 terms):
For the sum of the first 7 terms:  
\[ S_7 = \frac{3 (1 - 3^7)}{1 - 3} = \frac{3 (1 - 2187)}{-2} = \frac{3 \times (-2186)}{-2} = 3279 \]

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Do you need further details on any of these calculations?

Here are 5 related questions for practice:
1. What is the 10th term in the sequence \( 5, 10, 20, 40, \dots \)?
2. How would you find the sum of the first 8 terms in the sequence \( 5, 10, 20, 40, \dots \)?
3. What is the 15th term in the sequence \( 3, 9, 27, 81, \dots \)?
4. How do you find the sum of the first 10 terms of a geometric sequence?
5. What would the common ratio be if the sequence were \( 4, 12, 36, 108, \dots \)?

**Tip:** In a geometric sequence, always look for the common ratio by dividing any term by its previous term!

Find the 8th term and the sum of the first 6 terms of the sequence 5, 10, 20, 40,... and the 9th term and sum of the first 7 terms of the sequence 3, 9, 27, 81,...

Learn how to solve geometric sequences, including finding the 8th and 9th terms, and summing the first 6 and 7 terms of sequences such as 5, 10, 20, 40,... and 3, 9, 27, 81,... with step-by-step solutions. Suitable for high school students.

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