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The image shows a geometric sequence problem where you are provided with the 3rd term (\(U_3 = 12\)) and the 6th term (\(U_6 = 96\)) of the sequence. The tasks are to find:

a) The first term (\(a\))  
b) The common ratio (\(r\))  
c) The 8th term (\(U_8\))  
d) The sum of the first 6 terms (\(S_6\))

Let's solve these step by step.

### Step 1: General form of a geometric sequence
The \(n\)-th term of a geometric sequence is given by:
\[
U_n = a \cdot r^{n-1}
\]

### Step 2: Use the given terms
We know the following:
- \(U_3 = a \cdot r^2 = 12\)
- \(U_6 = a \cdot r^5 = 96\)

We now have two equations:
\[
a \cdot r^2 = 12 \quad \text{(1)}
\]
\[
a \cdot r^5 = 96 \quad \text{(2)}
\]

### Step 3: Solve for \(r\)
Divide equation (2) by equation (1):
\[
\frac{a \cdot r^5}{a \cdot r^2} = \frac{96}{12}
\]
This simplifies to:
\[
r^3 = 8 \quad \Rightarrow \quad r = 2
\]

### Step 4: Find \(a\)
Substitute \(r = 2\) into equation (1):
\[
a \cdot 2^2 = 12 \quad \Rightarrow \quad a \cdot 4 = 12 \quad \Rightarrow \quad a = 3
\]

### Step 5: Find the 8th term (\(U_8\))
Using the formula for the \(n\)-th term:
\[
U_8 = a \cdot r^{7} = 3 \cdot 2^7 = 3 \cdot 128 = 384
\]

### Step 6: Find the sum of the first 6 terms (\(S_6\))
The sum of the first \(n\) terms of a geometric sequence is given by:
\[
S_n = a \cdot \frac{1 - r^n}{1 - r}
\]
For \(S_6\):
\[
S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2} = 3 \cdot \frac{1 - 64}{-1} = 3 \cdot 63 = 189
\]

### Final Answers:
a) The first term \(a = 3\)  
b) The common ratio \(r = 2\)  
c) The 8th term \(U_8 = 384\)  
d) The sum of the first 6 terms \(S_6 = 189\)

Would you like further details on any part of the solution or have any questions?

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Here are 5 related questions to deepen understanding:

1. How do you calculate the sum of an infinite geometric series?
2. What would the 10th term of this geometric sequence be?
3. If the ratio was \(r = 3\), how would the sequence change?
4. How can you find the common ratio if you're given non-consecutive terms?
5. What is the difference between an arithmetic sequence and a geometric sequence?

**Tip**: In a geometric sequence, the ratio between consecutive terms is always constant, which helps in solving problems when some terms are unknown.

Given a geometric sequence with U3 = 12 and U6 = 96, find the first term, common ratio, 8th term, and sum of the first 6 terms.

Solve a geometric sequence problem where U3 = 12 and U6 = 96. Learn how to find the first term, common ratio, 8th term, and sum of the first 6 terms using key geometric formulas. Suitable for students in Grades 9-11.

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