Tell whether the sequence is geometric. Explain why or why not.
1. 4, 16, 64, 256, 1024,...
2. -1/4, 3/8, -3/16, 1/32, -3/64, ...
3. 0.75, 1.5, 2.25, 3, 3.75, ...
1. yes each term is 4 times the previous
2. 3/8 / -1/4 is not -3/16/ 3/8 so no
3. no for the same reason
To determine whether a sequence is geometric, we need to check if there is a common ratio between consecutive terms. If there is a constant ratio, the sequence is geometric; otherwise, it is not.
1. The sequence 4, 16, 64, 256, 1024,... can be written as 4^1, 4^2, 4^3, 4^4, 4^5,... This sequence has a common ratio of 4 because each term is obtained by multiplying the preceding term by 4. Therefore, the sequence is geometric.
2. The sequence -1/4, 3/8, -3/16, 1/32, -3/64,... does not have a constant ratio between consecutive terms. The sequence alternates between multiplying by -2 and dividing by 2. Therefore, the sequence is not geometric.
3. The sequence 0.75, 1.5, 2.25, 3, 3.75,... can be obtained by multiplying each term by 1.5. Each term is obtained by adding 0.75 to the preceding term. Therefore, the sequence is not geometric.