Divide consecutive terms:

$r = \frac{15}{5} = 3$

Check: $\frac{45}{15} = 3$ ✓

Answer: Common ratio = $3$

Step 1: Identify $a_1$ and $r$ $a_1 = 2, \quad r = \frac{6}{2} = 3$

Step 2: Use the explicit formula $a_n = a_1 \cdot r^{n-1}$

Step 3: Substitute $n = 8$ $a_8 = 2 \cdot 3^{8-1}$ $= 2 \cdot 3^7$ $= 2 \cdot 2187$ $= 4374$

Answer: $a_8 = 4374$

Given: $a_3 = 12$ and $a_6 = 96$

Step 1: Write equations using $a_n = a_1 \cdot r^{n-1}$ $a_3: \quad 12 = a_1 \cdot r^2$ $a_6: \quad 96 = a_1 \cdot r^5$

Step 2: Divide the second by the first $\frac{96}{12} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2}$ $8 = r^3$ $r = 2$

Step 3: Find $a_1$ using $a_3 = 12$ $12 = a_1 \cdot 2^2$ $12 = 4a_1$ $a_1 = 3$

Check: Sequence is $3, 6, 12, 24, 48, 96, ...$ ✓

Answer: $a_1 = 3$, $r = 2$