Question

hunter uses a running total to keep track of his score in a card game, and he notices that his score follows an arithmetic sequence. which of these sequences could be hunters scores?
show your work here
\\(\boldsymbol{\circ}\\) \\(-1, -8, -27, -64, -125, \dots\\) \\(\boldsymbol{\circ}\\) \\(-5, -2, 3, 10, 19, \dots\\)
\\(\boldsymbol{\circ}\\) \\(-5, -1, 3, 7, 11, \dots\\) \\(\boldsymbol{\circ}\\) \\(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots\\)

Explanation:

Step1: Recall Arithmetic Sequence Definition

An arithmetic sequence has a common difference \(d\), where \(d = a_{n + 1}-a_n\) for all \(n\). We check each sequence.

Step2: Check First Sequence \(-1, -8, -27, -64, -125, \dots\)

Calculate differences: \(-8 - (-1)=-7\), \(-27 - (-8)=-19\). Differences are not equal. Not arithmetic.

Step3: Check Second Sequence \(-5, -2, 3, 10, 19, \dots\)

Differences: \(-2 - (-5)=3\), \(3 - (-2)=5\). Differences not equal. Not arithmetic.

Step4: Check Third Sequence \(-5, -1, 3, 7, 11, \dots\)

Differences: \(-1 - (-5)=4\), \(3 - (-1)=4\), \(7 - 3 = 4\), \(11 - 7 = 4\). Common difference \(d = 4\). Arithmetic.

Step5: Check Fourth Sequence \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \dots\)

Differences: \(\frac{1}{4}-\frac{1}{2}=-\frac{1}{4}\), \(\frac{1}{8}-\frac{1}{4}=-\frac{1}{8}\). Differences not equal (geometric sequence). Not arithmetic.

Answer:

\(\boldsymbol{-5, -1, 3, 7, 11, \dots}\) (the third option)