Question

functions and lines
identifying arithmetic sequences and finding the common difference
for each sequence, determine whether it appears to be arithmetic.
if it does, find the common difference.
-5, -1, 3, 7, ...
○ arithmetic
common difference: d =
○ not arithmetic
-3, -6, -12, -24, ...
○ arithmetic
common difference: d =
○ not arithmetic
16, 20, 24, 28, ...
○ arithmetic
common difference: d =
○ not arithmetic

Explanation:

Step1: Analyze first sequence

The sequence is \(-5, -1, 3, 7, \dots\). To check if it's arithmetic, find the difference between consecutive terms.
\(-1 - (-5) = -1 + 5 = 4\)
\(3 - (-1) = 3 + 1 = 4\)
\(7 - 3 = 4\)
Since the differences are equal, it's arithmetic with \(d = 4\).

Step2: Analyze second sequence

The sequence is \(-3, -6, -12, -24, \dots\). Find the differences:
\(-6 - (-3) = -6 + 3 = -3\)
\(-12 - (-6) = -12 + 6 = -6\)
The differences \(-3\) and \(-6\) are not equal, so it's not arithmetic.

Step3: Analyze third sequence

The sequence is \(16, 20, 24, 28, \dots\). Find the differences:
\(20 - 16 = 4\)
\(24 - 20 = 4\)
\(28 - 24 = 4\)
Since the differences are equal, it's arithmetic with \(d = 4\).

Answer:

For \(-5, -1, 3, 7, \dots\): Arithmetic, \(d = 4\)
For \(-3, -6, -12, -24, \dots\): Not arithmetic
For \(16, 20, 24, 28, \dots\): Arithmetic, \(d = 4\)