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Recognizing Arithmetic Sequences

Recognizing Arithmetic Sequences 1.13 - Solution

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By observing the change that occurs between each consecutive term, we can describe the pattern of the sequence. Here, we see that the common difference from one term to the next is adding{\color{#0000FF}{\text{adding}}} 12.{\color{#0000FF}{\frac{1}{2}}}. -12+120+1212+121\begin{aligned} \text{-}\dfrac{1}{2}\stackrel{{\color{#0000FF}{+\tfrac{1}{2}}}}{\longrightarrow}0\stackrel{{\color{#0000FF}{+\tfrac{1}{2}}}}{\longrightarrow}\dfrac{1}{2}\stackrel{+{\color{#0000FF}{\tfrac{1}{2}}}}{\longrightarrow}1 \end{aligned} To find the next three terms in the sequence, we will extend this pattern three times. -12+120+1212+121+12112+122+12212\begin{aligned} \text{-}\dfrac{1}{2}\stackrel{+{\color{#0000FF}{\tfrac{1}{2}}}}{\longrightarrow}0\stackrel{+{\color{#0000FF}{\tfrac{1}{2}}}}{\longrightarrow}\dfrac{1}{2}\stackrel{+{\color{#0000FF}{\tfrac{1}{2}}}}{\longrightarrow}1\stackrel{+{\color{#0000FF}{\tfrac{1}{2}}}}{\longrightarrow} {\color{#0000FF}{1 \frac{1}{2}}}\stackrel{+{\color{#0000FF}{\tfrac{1}{2}}}}{\longrightarrow}{\color{#0000FF}{2}}\stackrel{{\color{#0000FF}{+\tfrac{1}{2}}}}{\longrightarrow}{\color{#0000FF}{2\frac{1}{2}}} \end{aligned}