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

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

## Want to know more about Polynomial arithmetic?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

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