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Answer :
Let's analyze and describe the pattern in each of the given sequences.
a. 2; 8; 32; 128; 512
This sequence is a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor. Here, the factor is 4.
[tex]2 \times 4 = 8 \\
8 \times 4 = 32 \\
32 \times 4 = 128 \\
128 \times 4 = 512 \\[/tex]
b. 4; 12; 36; 108; 324
This is also a geometric sequence. Each term is obtained by multiplying the previous term by 3.
[tex]4 \times 3 = 12 \\
12 \times 3 = 36 \\
36 \times 3 = 108 \\
108 \times 3 = 324 \\[/tex]
c. 6; 12; 24; 48; 96
This is another geometric sequence where each term is obtained by multiplying the previous term by 2.
[tex]6 \times 2 = 12 \\
12 \times 2 = 24 \\
24 \times 2 = 48 \\
48 \times 2 = 96 \\[/tex]
d. 8; 40; 200; 1000; 5000
This sequence is geometric with a factor of 5.
[tex]8 \times 5 = 40 \\
40 \times 5 = 200 \\
200 \times 5 = 1000 \\
1000 \times 5 = 5000 \\[/tex]
e. 1; 6; 36; 216; 1296
This sequence is geometric with a factor of 6.
[tex]1 \times 6 = 6 \\
6 \times 6 = 36 \\
36 \times 6 = 216 \\
216 \times 6 = 1296 \\[/tex]
f. 3; 9; 27; 81; 243
This is a geometric sequence with a factor of 3.
[tex]3 \times 3 = 9 \\
9 \times 3 = 27 \\
27 \times 3 = 81 \\
81 \times 3 = 243 \\[/tex]
g. 5; 20; 80; 320; 1280
This sequence is geometric with a factor of 4.
[tex]5 \times 4 = 20 \\
20 \times 4 = 80 \\
80 \times 4 = 320 \\
320 \times 4 = 1280 \\[/tex]
h. 7; 42; 252; 1512
This sequence is also geometric, and each term is multiplied by 6.
[tex]7 \times 6 = 42 \\
42 \times 6 = 252 \\
252 \times 6 = 1512 \\[/tex]
i. 9; 45; 225; 1125
In this sequence, each term is multiplied by 5.
[tex]9 \times 5 = 45 \\
45 \times 5 = 225 \\
225 \times 5 = 1125 \\[/tex]
j. 20; 40; 80; 160
This is the last geometric sequence, and each term is obtained by multiplying the previous one by 2.
[tex]20 \times 2 = 40 \\
40 \times 2 = 80 \\
80 \times 2 = 160 \\[/tex]
In summary, all these sequences exhibit a geometric progression where each term is a constant multiple of the previous one, characterized by the respective multiplying factor.
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