Question

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,  ...

Polynomial arithmetic
ANSWERED
asked 2020-11-22
For each sequence, decide whether it could be arithmetic,
\(a)  25,  5,  1,  ...\)
\(b)  25,  19,  13,  ...\)
\(c)  4,  9,  16,  ...\)
\(d)  50,  60,  70,  ...\)
\(e)  \frac{1}{2},  3,  18,  ...\)

Answers (1)

2020-11-23
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
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