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?

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.