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What kind of sequence is this? $\newline$ $2, 14, 98, 686, \ldots$ $\newline$ 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?

\newline

2, 14, 98, 686, \ldots

\newline

Choices:

\newline

(A) arithmetic

\newline

(B) geometric

\newline

(C) both

\newline

(D) neither

Identify given sequence: Identify the given sequence. $\newline$ The given sequence is $2, 14, 98, 686, \ldots$
Check arithmetic: Check if the sequence is arithmetic by finding the difference between consecutive terms. $\newline$ Difference between second and first term: $14 - 2 = 12$ $\newline$ Difference between third and second term: $98 - 14 = 84$ $\newline$ Difference between fourth and third term: $686 - 98 = 588$ $\newline$ The differences are not the same, so the sequence is not arithmetic.
Check geometric: Check if the sequence is geometric by finding the ratio between consecutive terms. $\newline$ Ratio of second to first term: $\frac{14}{2} = 7$ $\newline$ Ratio of third to second term: $\frac{98}{14} = 7$ $\newline$ Ratio of fourth to third term: $\frac{686}{98} = 7$ $\newline$ The ratios are the same, so the sequence is geometric.
Determine sequence type: Determine the type of sequence based on the findings from steps $2$ and $3$ . Since the sequence has a common ratio but not a common difference, it is a geometric sequence.

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