1. For each function below, determine whether it is discrete or not. Then, explain your choice. (-2.4) (a) a) Discrete? Y /N Explanation:
1. For each function below, determine whether it is discrete or not. Then, explain your choice. (-2.4) (a) a) Discrete? Y /N Explanation:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![1. For each function below, determine whether it is discrete or not. Then, explain your choice.
(6.7)
(-2.4)
(a, 5)
a)
Discrete? Y / N
Explanation:
b)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72575b8d-a22c-4c9f-ac91-435137a24931%2F4eb3e27f-b7f6-45f4-bbd1-803694fecf21%2F0mt7gf5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. For each function below, determine whether it is discrete or not. Then, explain your choice.
(6.7)
(-2.4)
(a, 5)
a)
Discrete? Y / N
Explanation:
b)
![Infinite and Finite Sequences
You may have noticed in 1b of the practice problems, that sequence had a . behind it. This means that the
sequence continues on, and if there is a pattern to that sequence, it does so following that pattern. From
this, we can make the two following statements:
1. A sequence that has no last term, but continues on indefinitely is an infinite sequence.
2. A sequence that has a last term is a finite sequence.
Here we encounter a problem. What if we wanted to write out an entire infinite sequence? By its very
definition, it would take us forever to do it! Take the sequence:
Clearly, if we kept counting, we could guess the next numbers on our list: 10, 12, 14, 16, etc. but since the
sequence goes on forever, we require a shorter way of writing it all out. After all, what if someone asked us
to find the 10 term in that sequence? We could just keep adding +2 until we reached it, but that's a lot of
unnecessary work.
Instead, if we figure out what rule / pattern the sequence follows as it goes from one term to the next, we
can come up with what is called a general term. The general term can be written out as t, where n can be
any number.
Ex. 1 Find the general term for the sequence 3, 6, 9, 12, .
To find the general term, let's first see if we can recognize a pattern:
Term Number
Term Value
S(n) or t,
1.
3
2
12
It looks like whatever our term number is, we can always multiply it by 3 to get out term value. Therefore,
the "rule" we're looking for is . If we write that out as an equation, then we get:
Let's test this rule out. According to the list above, the third number is 9, meaning when, should equal
Using our formula from above, we can substitute 3 for n, and if our math is right, we should get 9 back:
+ Based on this, our "rule" is correct.
Ex. 2 Find the general term for the sequence 1, 8, 27, 64,..
Again, to find the general term let's first see if we can recognize a pattern by asking ourselves what we can
do to "n" in order to change it into "t.":
Term Number
Term Value
S(n) or t,
3
27
64
It seems like we can take any of our n values, raise them to the exponent 3, and then get back a t, value. For
instance, 2' = 8 and 4' = 64. Therefore, our "rule" is:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F72575b8d-a22c-4c9f-ac91-435137a24931%2F4eb3e27f-b7f6-45f4-bbd1-803694fecf21%2Fh1d2vmm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Infinite and Finite Sequences
You may have noticed in 1b of the practice problems, that sequence had a . behind it. This means that the
sequence continues on, and if there is a pattern to that sequence, it does so following that pattern. From
this, we can make the two following statements:
1. A sequence that has no last term, but continues on indefinitely is an infinite sequence.
2. A sequence that has a last term is a finite sequence.
Here we encounter a problem. What if we wanted to write out an entire infinite sequence? By its very
definition, it would take us forever to do it! Take the sequence:
Clearly, if we kept counting, we could guess the next numbers on our list: 10, 12, 14, 16, etc. but since the
sequence goes on forever, we require a shorter way of writing it all out. After all, what if someone asked us
to find the 10 term in that sequence? We could just keep adding +2 until we reached it, but that's a lot of
unnecessary work.
Instead, if we figure out what rule / pattern the sequence follows as it goes from one term to the next, we
can come up with what is called a general term. The general term can be written out as t, where n can be
any number.
Ex. 1 Find the general term for the sequence 3, 6, 9, 12, .
To find the general term, let's first see if we can recognize a pattern:
Term Number
Term Value
S(n) or t,
1.
3
2
12
It looks like whatever our term number is, we can always multiply it by 3 to get out term value. Therefore,
the "rule" we're looking for is . If we write that out as an equation, then we get:
Let's test this rule out. According to the list above, the third number is 9, meaning when, should equal
Using our formula from above, we can substitute 3 for n, and if our math is right, we should get 9 back:
+ Based on this, our "rule" is correct.
Ex. 2 Find the general term for the sequence 1, 8, 27, 64,..
Again, to find the general term let's first see if we can recognize a pattern by asking ourselves what we can
do to "n" in order to change it into "t.":
Term Number
Term Value
S(n) or t,
3
27
64
It seems like we can take any of our n values, raise them to the exponent 3, and then get back a t, value. For
instance, 2' = 8 and 4' = 64. Therefore, our "rule" is:
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