Dillard

2020-11-22

For each sequence, decide whether it could be arithmetic,
$a\right) 25, 5, 1, ...$
$b\right) 25, 19, 13, ...$
$c\right) 4, 9, 16, ...$
$d\right) 50, 60, 70, ...$
$e\right) \frac{1}{2}, 3, 18, ...$

Step 1 Given:
$a\right) 25, 5, 1, ...b\right) 25, 19, 13, ...c\right) 4, 9, 16, ...d\right) 50, 60, 70, ...e\right) \frac{1}{2}, 3, 18, ...$
Step 2 Concept:
Let, the series ${a}_{1}, {a}_{2}, {a}_{3}, ...$
When
$d={a}_{2}-{a}_{1}$
$d={a}_{3}-{a}_{2}$
Then, the series is known as Arithmetic Series and d is called the common difference
When
$r=\frac{{a}_{2}}{{a}_{1}}$
$r=\frac{{a}_{3}}{{a}_{2}}$
Then, the series is known as the Geometric Series and r is called the common ratio
Part a
$25, 5, 1, ...$
Here, $r=\frac{{a}_{2}}{{a}_{1}}=\frac{25}{5}=5$
$r=\frac{{a}_{3}}{{a}_{2}}=\frac{5}{1}=5$
This series is a Geometric Series with a common ratio of 5
Part b
$25, 19, 13, ...$
Here, $d={a}_{2}-{a}_{1}=19-25=-6$
$d={a}_{3}-{a}_{2}=13-19=-6$
This series is Arithmetic Series with common difference –6
Part c
$4, 9, 16, ...$
Here, $d={a}_{2}-{a}_{1}=9-4=5$
$d={a}_{3}-{a}_{2}=16-9=7$
This series is not Arithmetic Series because the common difference is not the same
Continuation from the last step:
$4, 9, 16, ...$
Here, $r=\frac{{a}_{2}}{{a}_{1}}=\frac{9}{4}$
$r=\left({a}_{3}\right)/\left({a}_{2}\right)=\frac{16}{9}$
This series is not Geometric Series because the common ratio is not the same
Part d
$50, 60, 70, ...$
Here, $d={a}_{2}-{a}_{1}=60-50=10$
$d={a}_{3}-{a}_{2}=70-60=10$
This series is Arithmetic Series with a common difference of 10
Part e
$12, 3, 18, ...$
Here, $r=\frac{{a}_{2}}{{a}_{1}}=\frac{3}{\frac{1}{2}}=6$
$r=\frac{{a}_{3}}{{a}_{2}}=\frac{18}{3}=6$
This series is a Geometric Series with a common ratio of 6

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