Complete each of the following sequences:
1) 2, 4, 8, __, 32, 64, 128, ...
2) 3, 17, 31, __, 59, 73, ...
3) -9/2, __, -18, -36, ...
4) 2/3, 2, 6, __
5) -2, 4, -8, __, -32, 64, -128, ...
Determine if the sequence is geometric. If it is, find the common ratio.
6) -2, -10, -50, -250, ...
7) -1, 6, -36, 216, ...
8) -1, -5, -25, -125, ...
9) -2, 10, -50, 250, ...
10) -3, -1, -1/3, -1/9, ...
Answer (2)
The sequences were completed by identifying their patterns, such as repeated multiplication or increasing differences. Most sequences were determined to be geometric, with common ratios calculated where applicable. Examples include sequences like -2, -10, -50 where each term is multiplied by 5.
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Let's complete and analyze each sequence step-by-step to determine the missing numbers and find if a sequence is geometric.
Sequence: 2, 4, 8, __, 32, 64, 128, ...
This is a geometric sequence because each term is obtained by multiplying the previous term by 2.
Next term: 8 × 2 = 16
Complete Sequence: 2, 4, 8, 16, 32, 64, 128...
Common Ratio: 2
Sequence: 3, 17, 31, __, 59, 73, ...
This is an arithmetic sequence where each term increases by 14.
Next term: 31 + 14 = 45
Complete Sequence: 3, 17, 31, 45, 59, 73...
Common Difference: 14
Sequence: -\frac{9}{2}, __, -18, -36, ...
It appears to follow a geometric pattern where each term is multiplied by 4.
Next term: − 2 9 × 4 = − 18
Complete Sequence: -\frac{9}{2}, -18, -36...
Common Ratio: 4
Sequence: \frac{2}{3}, 2, 6, __
This sequence multiplies by 3, thus it's geometric.
Next term: 6 × 3 = 18
Complete Sequence: \frac{2}{3}, 2, 6, 18...
Common Ratio: 3
Sequence: -2, 4, -8, __, -32, 64, -128, ...
This is geometric as each term is multiplied by -2.
Next term: − 8 × ( − 2 ) = 16
Complete Sequence: -2, 4, -8, 16, -32, 64, -128...
Common Ratio: -2
Now, let's determine if the following sequences are geometric and find their common ratio if they are:
Sequence: -2, -10, -50, -250, ...
This is geometric; each term is multiplied by 5.
Common Ratio: 5
Sequence: -1, 6, -36, 216, ...
This is geometric; each term is multiplied by -6.
Common Ratio: -6
Sequence: -1, -5, -25, -125, ...
This is geometric; each term is multiplied by 5.
Common Ratio: 5
Sequence: -2, 10, -50, 250, ...
This is geometric; each term is multiplied by -5.
Common Ratio: -5
Sequence: -3, -1, -13, -19, ...
This sequence is not geometric because it does not have a constant ratio between the terms.
