Meg is writing a poem. She writes $1$ word on the first line, $2$ words on the second line, $4$ words on the third line, and $8$ words on the fourth line. What kind of sequence is this? $\newline$ Choices: $\newline$ (A) arithmetic $\newline$ (B) geometric $\newline$ (C) both $\newline$ (D) neither

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Grade 8

Identify arithmetic and geometric sequences

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Q. Meg is writing a poem. She writes

1

word on the first line,

2

words on the second line,

4

words on the third line, and

8

words on the fourth line. What kind of sequence is this?

\newline

Choices:

\newline

(A) arithmetic

\newline

(B) geometric

\newline

\newline

(D) neither

Identify Sequence Type: To determine the type of sequence, we need to look at the pattern of the word counts on each line. The word counts are as follows: $1, 2, 4, 8$ .
Check Arithmetic Sequence: An arithmetic sequence is one where each term after the first is obtained by adding a constant, called the common difference, to the previous term. Let's check if this is an arithmetic sequence by finding the difference between consecutive terms.
Check Geometric Sequence: The difference between the first and second term is $2 - 1 = 1$ . The difference between the second and third term is $4 - 2 = 2$ . Since the differences are not the same, this is not an arithmetic sequence.
Check Geometric Sequence: The difference between the first and second term is $2 - 1 = 1$ . The difference between the second and third term is $4 - 2 = 2$ . Since the differences are not the same, this is not an arithmetic sequence.A geometric sequence is one where each term after the first is obtained by multiplying the previous term by a constant, called the common ratio. Let's check if this is a geometric sequence by finding the ratio between consecutive terms.
Check Geometric Sequence: The difference between the first and second term is $2 - 1 = 1$ . The difference between the second and third term is $4 - 2 = 2$ . Since the differences are not the same, this is not an arithmetic sequence.A geometric sequence is one where each term after the first is obtained by multiplying the previous term by a constant, called the common ratio. Let's check if this is a geometric sequence by finding the ratio between consecutive terms.The ratio between the first and second term is $\frac{2}{1} = 2$ . The ratio between the second and third term is $\frac{4}{2} = 2$ . The ratio between the third and fourth term is $\frac{8}{4} = 2$ . Since the ratios are the same, this is a geometric sequence.