Sweet Time Bakery just opened and is increasing the number of items they bake. For example, the bakery made $36$ carrot cakes in December, $49$ carrot cakes in January, $64$ carrot cakes in February, and $81$ carrot cakes in March. What kind of sequence is this? $\newline$ Choices: $\newline$ (A) arithmetic $\newline$ (B) geometric $\newline$ (C) both $\newline$ (D) neither

Full solution

Q. Sweet Time Bakery just opened and is increasing the number of items they bake. For example, the bakery made

36

carrot cakes in December,

49

carrot cakes in January,

64

carrot cakes in February, and

81

carrot cakes in March. What kind of sequence is this?

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Choices:

\newline

(A) arithmetic

\newline

(B) geometric

\newline

\newline

(D) neither

Arithmetic Sequence Check: Let's examine the differences between the numbers of cakes made in consecutive months to determine if it's an arithmetic sequence. $\newline$ Arithmetic sequence check: $\newline$ January - December: $49 - 36 = 13$ $\newline$ February - January: $64 - 49 = 15$ $\newline$ March - February: $81 - 64 = 17$ $\newline$ The differences are not constant, so it is not an arithmetic sequence.
Geometric Sequence Check: Now let's check if it's a geometric sequence by examining the ratios of the numbers of cakes made in consecutive months. $\newline$ Geometric sequence check: $\newline$ January / December: $\frac{49}{36}$ $\newline$ February / January: $\frac{64}{49}$ $\newline$ March / February: $\frac{81}{64}$ $\newline$ Calculate the ratios to see if they are constant. $\newline$ $\frac{49}{36} = 1.3611$ (rounded to $4$ decimal places) $\newline$ $\frac{64}{49} = 1.3061$ (rounded to $4$ decimal places) $\newline$ $\frac{81}{64} = 1.2656$ (rounded to $4$ decimal places) $\newline$ The ratios are not constant, so it is not a geometric sequence.
Neither: Since the sequence is neither arithmetic (constant differences) nor geometric (constant ratios), the correct choice is: $\newline$ (D) neither