Step 1 Given:

\(a) 25, 5, 1, ...b) 25, 19, 13, ...c) 4, 9, 16, ...d) 50, 60, 70, ...e) \frac{1}{2}, 3, 18, ...\)

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, \(r=\frac{a_{2}}{a_{1}}=\frac{9}{4}\)

\(r=(a_{3})/(a_{2}) = \frac{16}{9}\)

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

\(a) 25, 5, 1, ...b) 25, 19, 13, ...c) 4, 9, 16, ...d) 50, 60, 70, ...e) \frac{1}{2}, 3, 18, ...\)

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, \(r=\frac{a_{2}}{a_{1}}=\frac{9}{4}\)

\(r=(a_{3})/(a_{2}) = \frac{16}{9}\)

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