Explain why a constant sequence is both arithmetic and geometric.
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Hi Ben,
Recall the definitions for arithmetic and geometric sequences. Arithmetic sequences require that the common difference between consecutive terms is constant. Geometric sequences, on the other hand, require that the common ratio between consecutive terms be constant.
So in your question, you have a constant sequence which means that the terms are unchanging. To consider whether the constant sequence is arithmetic, calculate the difference between any successive terms: a - a = 0. Because that will be the difference between any two consecutive terms, there is a common difference between consecutive terms, and the constant sequence is arithmetic.
To consider whether the constant sequence is geometric, calculate the ratio between any successive terms: a/a = 1 (assuming a != 0). Because this is ratio of any two consecutive terms, there is a common ratio between consecutive terms, and the constant sequence is geometric, provided a != 0.
To help reinforce your understanding of these concepts, I've searched and found a webpage and a video tutorial that address problems similar to this one, and I thought they might be helpful to you. I've listed them below.
As always, if you need more help, please clarify where you are in the process and what's giving you trouble. I'd be more than happy to continue to assist.
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Recall the definitions for arithmetic and geometric sequences. Arithmetic sequences require that the common difference between consecutive terms is constant. Geometric sequences, on the other hand, require that the common ratio between consecutive terms be constant.
So in your question, you have a constant sequence which means that the terms are unchanging. To consider whether the constant sequence is arithmetic, calculate the difference between any successive terms: a - a = 0. Because that will be the difference between any two consecutive terms, there is a common difference between consecutive terms, and the constant sequence is arithmetic.
To consider whether the constant sequence is geometric, calculate the ratio between any successive terms: a/a = 1 (assuming a != 0). Because this is ratio of any two consecutive terms, there is a common ratio between consecutive terms, and the constant sequence is geometric, provided a != 0.
To help reinforce your understanding of these concepts, I've searched and found a webpage and a video tutorial that address problems similar to this one, and I thought they might be helpful to you. I've listed them below.
As always, if you need more help, please clarify where you are in the process and what's giving you trouble. I'd be more than happy to continue to assist.
If we've been helpful in answering your question, consider stopping by our Facebook page at www.facebook.com/protutorcompany.
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in the case of an arithmetic sequence the constant difference d =0 , in the case of geometric sequence the constant ratio r would be equal to 1