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While counting chairs in the lunch room, Rachel noticed that there were $3$ chairs in the first row, $9$ chairs in the second row, $27$ chairs in the third row, and $81$ chairs in the fourth row. What kind of sequence is this? $\newline$ 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. While counting chairs in the lunch room, Rachel noticed that there were

3

chairs in the first row,

9

chairs in the second row,

27

chairs in the third row, and

81

chairs in the fourth row. What kind of sequence is this?

\newline

Choices:

\newline

(A) arithmetic

\newline

(B) geometric

\newline

(C) both

\newline

(D) neither

Count Chairs: Step $1$ : Rachel counted chairs in different rows. $\newline$ First row: $3$ chairs $\newline$ Second row: $9$ chairs $\newline$ Third row: $27$ chairs $\newline$ Fourth row: $81$ chairs $\newline$ List the number of chairs in the given order. $\newline$ Sequence: $3, 9, 27, 81$
Check Ratios: Step $2$ : $3, 9, 27, 81$ $\newline$ Are the ratios between consecutive terms equal? $\newline$ $\frac{9}{3} = 3$ $\newline$ $\frac{27}{9} = 3$ $\newline$ $\frac{81}{27} = 3$ $\newline$ The ratio between each consecutive term is the same, which is $3$ .
Identify Sequence: Step $3$ : Since the ratio between consecutive terms is constant, the sequence is geometric.

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