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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 =

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 =

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
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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, =
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:
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