Question

match the sequence to its common difference.
5, 8, 11, 14, ...
886, 855, 824, 793, ...
-46, -37, -28, -19, ...
2, -9, -20, -31, ...
14.2, 17.9, 21.6, 25.3, ...
-31
3
3.7
\frac{1}{3}
-11
\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, ...
0

Explanation:

Step1: Recall common difference formula

The common difference \(d\) of an arithmetic sequence is \(d = a_{n+1}-a_n\), where \(a_{n+1}\) and \(a_n\) are consecutive terms.

Step2: Analyze sequence \(5, 8, 11, 14, \dots\)

Calculate \(d = 8 - 5 = 3\), so common difference is \(3\).

Step3: Analyze sequence \(2, -9, -20, -31, \dots\)

Calculate \(d=-9 - 2=-11\), so common difference is \(-11\).

Step4: Analyze sequence \(0\) (assuming it's a sequence with one term, or maybe a typo; but if we consider a sequence like \(0, 0, 0, \dots\), \(d = 0 - 0 = 0\))

Step5: Analyze sequence \(14.2, 17.9, 21.6, 25.3, \dots\)

Calculate \(d = 17.9 - 14.2 = 3.7\), so common difference is \(3.7\).

Step6: Analyze sequence \(-46, -37, -28, -19, \dots\)

Calculate \(d=-37-(-46)=9\), so common difference is \(9\).

Step7: Analyze sequence \(886, 855, 824, 793, \dots\)

Calculate \(d = 855 - 886=-31\), so common difference is \(-31\).

Step8: Analyze sequence \(\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, \dots\)

First, rewrite \(3\) as \(\frac{9}{3}\). Then \(d=\frac{9}{3}-\frac{10}{3}=-\frac{1}{3}\)? Wait, no: \(\frac{8}{3}-\frac{9}{3}=-\frac{1}{3}\), but wait, \(3-\frac{10}{3}=\frac{9 - 10}{3}=-\frac{1}{3}\), and \(\frac{8}{3}-3=\frac{8 - 9}{3}=-\frac{1}{3}\), \(\frac{7}{3}-\frac{8}{3}=-\frac{1}{3}\). Wait, but the common difference here is \(-\frac{1}{3}\)? Wait, maybe I made a mistake. Wait, the problem has a box with \(\frac{1}{3}\), maybe a sign error? Wait, no, let's check again: \(\frac{10}{3}, 3=\frac{9}{3}, \frac{8}{3}, \frac{7}{3}\). So \(d=\frac{9}{3}-\frac{10}{3}=-\frac{1}{3}\), but if we reverse, \(\frac{10}{3}-3=\frac{1}{3}\). Wait, maybe the sequence is written in reverse? If the sequence is \(\frac{7}{3}, \frac{8}{3}, 3, \frac{10}{3}\), then \(d=\frac{1}{3}\). So common difference is \(\frac{1}{3}\) (if we consider the sequence in increasing order from \(\frac{7}{3}\) to \(\frac{10}{3}\)).

Now, matching:

• \(5, 8, 11, 14, \dots\) ↔ \(3\)
• \(2, -9, -20, -31, \dots\) ↔ \(-11\)
• \(0\) ↔ \(0\)
• \(14.2, 17.9, 21.6, 25.3, \dots\) ↔ \(3.7\)
• \(-46, -37, -28, -19, \dots\) ↔ \(9\)
• \(886, 855, 824, 793, \dots\) ↔ \(-31\)
• \(\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, \dots\) ↔ \(-\frac{1}{3}\)? Wait, no, maybe the common difference is \(-\frac{1}{3}\), but the box has \(\frac{1}{3}\). Wait, maybe I messed up. Wait, \(3-\frac{10}{3}=\frac{9 - 10}{3}=-\frac{1}{3}\), \(\frac{8}{3}-3=\frac{8 - 9}{3}=-\frac{1}{3}\), so common difference is \(-\frac{1}{3}\), but if the box is \(\frac{1}{3}\), maybe the sequence is \(\frac{7}{3}, \frac{8}{3}, 3, \frac{10}{3}\), then \(d=\frac{1}{3}\). So that sequence matches \(\frac{1}{3}\).

Answer:

• Sequence \(5, 8, 11, 14, \dots\) ↔ Common Difference \(3\)
• Sequence \(2, -9, -20, -31, \dots\) ↔ Common Difference \(-11\)
• Sequence \(0\) ↔ Common Difference \(0\)
• Sequence \(14.2, 17.9, 21.6, 25.3, \dots\) ↔ Common Difference \(3.7\)
• Sequence \(-46, -37, -28, -19, \dots\) ↔ Common Difference \(9\)
• Sequence \(886, 855, 824, 793, \dots\) ↔ Common Difference \(-31\)
• Sequence \(\frac{10}{3}, 3, \frac{8}{3}, \frac{7}{3}, \dots\) ↔ Common Difference \(-\frac{1}{3}\) (or \(\frac{1}{3}\) if sequence is reversed)