Question

5, 8, 11, 14, ...
2, -9, -20, -31, ...
14.2, 17.9, 21.6, 25.3, ...
886, 855, 824, 793, ...
-46, -37, -28, -19, ...
\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, ...
the sequences displayed to the left are examples of arithmetic sequences.
what do you notice? what do you wonder?
i notice ...
i wonder ...

Explanation:

Step1: Analyze the first sequence \(5, 8, 11, 14, \dots\)

Calculate the difference between consecutive terms: \(8 - 5 = 3\), \(11 - 8 = 3\), \(14 - 11 = 3\). So the common difference \(d = 3\).

Step2: Analyze the second sequence \(2, -9, -20, -31, \dots\)

Find the differences: \(-9 - 2 = -11\), \(-20 - (-9) = -11\), \(-31 - (-20) = -11\). Common difference \(d = -11\).

Step3: Analyze the third sequence \(14.2, 17.9, 21.6, 25.3, \dots\)

Differences: \(17.9 - 14.2 = 3.7\), \(21.6 - 17.9 = 3.7\), \(25.3 - 21.6 = 3.7\). Common difference \(d = 3.7\).

Step4: Analyze the fourth sequence \(886, 855, 824, 793, \dots\)

Differences: \(855 - 886 = -31\), \(824 - 855 = -31\), \(793 - 824 = -31\). Common difference \(d = -31\).

Step5: Analyze the fifth sequence \(-46, -37, -28, -19, \dots\)

Differences: \(-37 - (-46) = 9\), \(-28 - (-37) = 9\), \(-19 - (-28) = 9\). Common difference \(d = 9\).

Step6: Analyze the sixth sequence \(\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, \dots\)

First, rewrite \(3\) as \(\frac{9}{3}\). Then differences: \(\frac{9}{3} - \frac{10}{3} = -\frac{1}{3}\), \(\frac{8}{3} - \frac{9}{3} = -\frac{1}{3}\), \(\frac{7}{3} - \frac{8}{3} = -\frac{1}{3}\). Common difference \(d = -\frac{1}{3}\).

Answer:

All the given sequences are arithmetic sequences because in each sequence, the difference between consecutive terms (common difference) is constant. For example:

• \(5, 8, 11, 14, \dots\) has \(d = 3\)
• \(2, -9, -20, -31, \dots\) has \(d = -11\)
• \(14.2, 17.9, 21.6, 25.3, \dots\) has \(d = 3.7\)
• \(886, 855, 824, 793, \dots\) has \(d = -31\)
• \(-46, -37, -28, -19, \dots\) has \(d = 9\)
• \(\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, \dots\) has \(d = -\frac{1}{3}\)