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What kind of sequence is this? $\newline$ $22, 66, 198, 594, \dots$ $\newline$ Choices: $\newline$ (A) arithmetic $\newline$ (B) geometric $\newline$ (C) both $\newline$ (D) neither

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Identify arithmetic and geometric sequences

Full solution

Q. What kind of sequence is this?

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22, 66, 198, 594, \dots

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Choices:

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(A) arithmetic

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(B) geometric

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(C) both

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(D) neither

Determine Sequence Type: To determine the type of sequence, we need to examine the relationship between consecutive terms.
Check Arithmetic Sequence: First, let's check if it's an arithmetic sequence by finding the difference between consecutive terms.
Calculate Differences: The difference between the second term $66$ and the first term $22$ is $66 - 22 = 44$ .
Check Geometric Sequence: The difference between the third term $198$ and the second term $66$ is $198 - 66 = 132$ .
Calculate Ratios: Since the differences are not the same ( $44$ and $132$ ), this is not an arithmetic sequence.
Calculate Ratios: Since the differences are not the same ( $44$ and $132$ ), this is not an arithmetic sequence.Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.
Calculate Ratios: Since the differences are not the same ( $44$ and $132$ ), this is not an arithmetic sequence.Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.The ratio between the second term ( $66$ ) and the first term ( $22$ ) is $66 / 22 = 3$ .
Calculate Ratios: Since the differences are not the same ( $44$ and $132$ ), this is not an arithmetic sequence.Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.The ratio between the second term ( $66$ ) and the first term ( $22$ ) is $66 / 22 = 3$ .The ratio between the third term ( $198$ ) and the second term ( $66$ ) is $198 / 66 = 3$ .
Calculate Ratios: Since the differences are not the same ( $44$ and $132$ ), this is not an arithmetic sequence.Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.The ratio between the second term ( $66$ ) and the first term ( $22$ ) is $\frac{66}{22} = 3$ .The ratio between the third term ( $198$ ) and the second term ( $66$ ) is $\frac{198}{66} = 3$ .The ratio between the fourth term ( $594$ ) and the third term ( $198$ ) is $132$ $0$ .
Calculate Ratios: Since the differences are not the same ( $44$ and $132$ ), this is not an arithmetic sequence.Now, let's check if it's a geometric sequence by finding the ratio between consecutive terms.The ratio between the second term ( $66$ ) and the first term ( $22$ ) is $66 / 22 = 3$ .The ratio between the third term ( $198$ ) and the second term ( $66$ ) is $198 / 66 = 3$ .The ratio between the fourth term ( $594$ ) and the third term ( $198$ ) is $132$ $0$ .Since the ratios are the same (all are $132$ $1$ ), this is a geometric sequence.

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