1. Find the common ratio of the geometric sequence: 3, 6, 12, 24, 48, ... - The common ratio $r$ is found by dividing any term by its previous term. - Calculate $r = \frac{6}{3} = 2$. 2. What is the 5th term of the geometric sequence: $a=2$, $r=4$? - The formula for the $n$th term is $a_n = a \cdot r^{n-1}$. - Substitute $n=5$: $a_5 = 2 \cdot 4^{4} = 2 \cdot 256 = 512$. 3. Find the 8th term of the geometric sequence: 5, 10, 20, 40, ... - Calculate common ratio: $r = \frac{10}{5} = 2$. - Use formula: $a_8 = 5 \cdot 2^{7} = 5 \cdot 128 = 640$. 4. Find the sum of the first 6 terms of the geometric sequence: $a=3$, $r=2$ - Sum formula for first $n$ terms: $S_n = a \frac{r^{n} - 1}{r - 1}$. - Calculate $S_6 = 3 \cdot \frac{2^{6} - 1}{2-1} = 3 \cdot \frac{64 -1}{1} = 3 \cdot 63 = 189$. 5. Find the sum of the first 5 terms of the geometric sequence: 2, 6, 18, 54, ... - Common ratio: $r = \frac{6}{2} = 3$. - $a=2$, $n=5$, sum $S_5 = 2 \cdot \frac{3^{5} -1}{3-1} = 2 \cdot \frac{243 -1}{2} = 2 \cdot 121 = 242$. 6. The 1st term of a geometric sequence is 100 and the ratio is 0.5. Find the sum of the first 8 terms. - $a=100$, $r=0.5$, $n=8$. - $S_8 = 100 \cdot \frac{0.5^{8} -1}{0.5-1} = 100 \cdot \frac{0.00390625 -1}{-0.5} = 100 \cdot \frac{-0.99609375}{-0.5} = 100 \cdot 1.9921875 = 199.21875$. 7. The 1st term of a geometric sequence is 81, and the 4th term is 3. Find the common ratio and the sum to infinity, if possible. - Use formula $a_4 = a \cdot r^{3} \Rightarrow 3 = 81 \cdot r^{3}$. - Solve for $r^{3} = \frac{3}{81} = \frac{1}{27}$. - Therefore, $r = \sqrt[3]{\frac{1}{27}} = \frac{1}{3}$. - Since $|r| = \frac{1}{3} < 1$, sum to infinity exists. - Sum to infinity, $S_\infty = \frac{a}{1-r} = \frac{81}{1 - \frac{1}{3}} = \frac{81}{\frac{2}{3}} = 81 \cdot \frac{3}{2} = 121.5$.