Question

functions and lines
identifying arithmetic sequences and finding the common difference
3/5
for each sequence, determine whether it appears to be arithmetic.
if it does, find the common difference.
-5, -8, -11, -14, ...
arithmetic
common difference: d =
not arithmetic
8, 15, 22, 30, ...
arithmetic
common difference: d =
not arithmetic
2, -6, 18, -54, ...
arithmetic
common difference: d =
not arithmetic

Explanation:

Step1: Analyze the first sequence

The sequence is \(-5, -8, -11, -14, \dots\). To check if it's arithmetic, we find the difference between consecutive terms.
\(-8 - (-5) = -8 + 5 = -3\)
\(-11 - (-8) = -11 + 8 = -3\)
\(-14 - (-11) = -14 + 11 = -3\)
Since the difference is constant (\(-3\)), it is arithmetic with common difference \(d = -3\).

Step2: Analyze the second sequence

The sequence is \(8, 15, 22, 30, \dots\). Find the differences:
\(15 - 8 = 7\)
\(22 - 15 = 7\)
\(30 - 22 = 8\)
The differences are \(7\) and \(8\), which are not constant. So it is not arithmetic.

Step3: Analyze the third sequence

The sequence is \(2, -6, 18, -54, \dots\). Find the differences:
\(-6 - 2 = -8\)
\(18 - (-6) = 24\)
\(-54 - 18 = -72\)
The differences are \(-8\), \(24\), \(-72\), which are not constant. So it is not arithmetic.

Answer:

1. For the sequence \(-5, -8, -11, -14, \dots\): Arithmetic, Common difference \(d = -3\)
2. For the sequence \(8, 15, 22, 30, \dots\): Not arithmetic
3. For the sequence \(2, -6, 18, -54, \dots\): Not arithmetic