Question

12.1 geometric sequences 31) geometric sequences have a common _____2) to find the common ratio $(r)=$ _____3) to find the nth term of a geometric sequence use the explicit formula:$a_n=a_1cdot r^{n-1}$determine if the sequence is geometric. if it is, find the common ratio.4) 8, 10, 13, 17, ...5) 4, 24, 144, 864, ...6) 2, -4, 8, -16, ...7) 4, 16, 36, 64, ...determine if the sequence is geometric. if it is, find the common ratio and the three terms in the sequence after the last one given.8) 4, 16, 36, 64, ...9) 3, 15, 75, 375, ...10) -2, 8, -32, 128, ...11) 2, 8, 32, 128, ...

Explanation:

Step1: Answer fill-in 1

Geometric sequences have a common ratio.

Step2: Answer fill-in 2

To find the common ratio $(r) = \frac{\text{any term}}{\text{previous term}}$

Step3: Verify sequence 4

Check ratios: $\frac{10}{8}=1.25$, $\frac{13}{10}=1.3$, $\frac{17}{13}\approx1.308$. Ratios are not equal.

Step4: Verify sequence 5

Check ratios: $\frac{24}{4}=6$, $\frac{144}{24}=6$, $\frac{864}{144}=6$. Ratios are equal.

Step5: Verify sequence 6

Check ratios: $\frac{-4}{2}=-2$, $\frac{8}{-4}=-2$, $\frac{-16}{8}=-2$. Ratios are equal.

Step6: Verify sequence 7

Check ratios: $\frac{16}{4}=4$, $\frac{36}{16}=2.25$, $\frac{64}{36}\approx1.778$. Ratios are not equal.

Step7: Verify sequence 8

Check ratios: $\frac{16}{4}=4$, $\frac{36}{16}=2.25$, $\frac{64}{36}\approx1.778$. Ratios are not equal.

Step8: Verify sequence 9

Check ratios: $\frac{15}{3}=5$, $\frac{75}{15}=5$, $\frac{375}{75}=5$. Ratios are equal. Find next terms: $375\times5=1875$, $1875\times5=9375$, $9375\times5=46875$.

Step9: Verify sequence 10

Check ratios: $\frac{8}{-2}=-4$, $\frac{-32}{8}=-4$, $\frac{128}{-32}=-4$. Ratios are equal. Find next terms: $128\times(-4)=-512$, $-512\times(-4)=2048$, $2048\times(-4)=-8192$.

Step10: Verify sequence 11

Check ratios: $\frac{8}{2}=4$, $\frac{32}{8}=4$, $\frac{128}{32}=4$. Ratios are equal. Find next terms: $128\times4=512$, $512\times4=2048$, $2048\times4=8192$.

Answer:

1. ratio
2. $\frac{\text{any term}}{\text{previous term}}$
3. Not a geometric sequence
4. Geometric sequence, common ratio $r=6$
5. Geometric sequence, common ratio $r=-2$
6. Not a geometric sequence
7. Not a geometric sequence
8. Geometric sequence, common ratio $r=5$; next three terms: 1875, 9375, 46875
9. Geometric sequence, common ratio $r=-4$; next three terms: -512, 2048, -8192
10. Geometric sequence, common ratio $r=4$; next three terms: 512, 2048, 8192