Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
4. The same strategy from the previous page can be used to find specific entries. Given the function notation formula for the nth term in a sequence, find t, t, and t,. a) t, = t, = t, = b) t, = t, = t, =

4. The same strategy from the previous page can be used to find specific entries. Given the function notation formula for the nth term in a sequence, find t, t, and t,. a) t, = t, = t, = b) t, = t, = t, =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
Pls give clear explanation, thank you!
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:
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:
4. The same strategy from the previous page can be used to find specific entries. Given the function notation formula for the nth term in a sequence, find t, t, and t,. a) t,= t = t, = b) t, = t = t, =
Transcribed Image Text:4. The same strategy from the previous page can be used to find specific entries. Given the function notation formula for the nth term in a sequence, find t, t, and t,. a) t,= t = t, = b) t, = t = t, =
Expert Solution
steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS