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While sorting some buttons, Keith put $49$ buttons in the first box, $64$ buttons in the second box, $81$ buttons in the third box, and $100$ buttons in the fourth box. 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 sorting some buttons, Keith put

49

buttons in the first box,

64

buttons in the second box,

81

buttons in the third box, and

100

buttons in the fourth box. What kind of sequence is this?

\newline

Choices:

\newline

(A) arithmetic

\newline

(B) geometric

\newline

(C) both

\newline

(D) neither

Identify Pattern: First, let's list the number of buttons in each box to see if we can identify a pattern: $\newline$ First box: $49$ buttons $\newline$ Second box: $64$ buttons $\newline$ Third box: $81$ buttons $\newline$ Fourth box: $100$ buttons
Check Arithmetic Sequence: To determine if this is an arithmetic sequence, we need to check if the difference between consecutive terms is constant. $\newline$ Difference between second and first box: $64 - 49 = 15$ $\newline$ Difference between third and second box: $81 - 64 = 17$ $\newline$ Difference between fourth and third box: $100 - 81 = 19$ $\newline$ Since the differences are not constant, this is not an arithmetic sequence.
Check Geometric Sequence: To determine if this is a geometric sequence, we need to check if the ratio between consecutive terms is constant. $\newline$ Ratio of second to first box: $\frac{64}{49}$ $\newline$ Ratio of third to second box: $\frac{81}{64}$ $\newline$ Ratio of fourth to third box: $\frac{100}{81}$ $\newline$ We can see that these ratios are not the same, so this is not a geometric sequence.
Final Conclusion: Since the sequence is neither arithmetic (constant difference) nor geometric (constant ratio), the correct choice is: $\newline$ (D) neither

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