Consecutive Numbers<\/p>\n
Think of three consecutive whole numbers. 1, 2, 3 or 45, 46, 47 or 201, 201, 203<\/p>\n
Think of three consecutive whole numbers bigger than a million: 2000000, 2000001, 2000002<\/p>\n
Think of three consecutive odd numbers: 7, 9, 11<\/p>\n
Think of three consecutive two-digit primes: 29, 31, 37 or 43, 47, 53<\/p>\n
Think of three consecutive perfect squares: 16, 25, 36 or 49, 64, 81<\/p>\n
Patterns<\/p>\n
A. 1, 10, 2, 20, 3, 30, 4, 40, 5, 50, …<\/p>\n
B. 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, …<\/p>\n
C. 1, 10, 100, 1000, 10000, 100000, …<\/p>\n
Can you think of an example of a geometric sequence where the terms get smaller and smaller (instead of larger and larger)? 1, 0.1, 0.01, 0.001, 0.0001, …<\/p>\n
(This can be written as 1, 1\/10, 1\/100, 1\/1000, …)<\/p>\n
What is another geometric sequence where the terms get smaller and smaller? Think of one. <\/p>\n
81, 9, 1, …<\/p>\n
What day of the week will it be in 1473 days? (If it is Wednesday today): Saturday<\/p>\n
Dr. Betty’s bacteria population doubles every 12 hours. She puts some bacteria in a bucket and several days later, on Friday at 11 AM, she has 4 gallons of bacteria. At what day and time does she have 2 gallons of bacteria? At what time does she have one quart of bacteria? (A quart is one-quarter of a gallon.) On Thursday, 11 PM she has 2 gallons. On Wednesday, 11 AM she has a quart. <\/p>\n
Choose any three consecutive terms in the Fibonacci sequence. Square the middle term and multiply the outer two terms together. What pattern do you find? The answer is either 1 or -1. This is called Cassini’s identity. <\/p>\n
Another way to generate sequences is by rules. It is interesting to see what pattern results from a set of rules. Here is an example. <\/p>\n
Rule 1: If the number is less than 10, add 3 to get the next term.Rule 3: If the number is equal to 10, subtract 5 to get the next term.<\/p>\n
Rule 4: If the number is greater than 10, subtract 6 to get the next term.<\/p>\n
Start with any number.<\/p>\n
If the first term is 9, what is the 100th term? 9<\/p>\n
If the first term is 2, what is the 100th term? 11<\/p>\n
Destinations<\/p>\n
Can you think of another sequence that goes to zero?
\n0.2, 0.02, 0.002, 0.0002, …<\/p>\n
or<\/p>\n
1, -1, 1\/2, -1\/2, 1\/3, -1\/3, …<\/p>\n
Can you think of another sequence that goes to infinity? 1, 3, 9, 27, …<\/p>\n
or<\/p>\n
2, 3, 5, 7, 11, 13, …<\/p>\n
1, 1.01, 1.02, …<\/p>\n
Think of an infinite sequence that converges to 10: 9, 9.9, 9.99, 9.999, …<\/p>\n
or<\/p>\n
10.1, 10.01, 10.001, …<\/p>\n
Think of an infinite sequence that converges to 10 in an unusual way: <\/p>\n
10 + 1\/2, 10 + 1\/3, 10 + 1\/5, 10 + 1\/7, 10 + 1\/11, 10 + 1\/13, …<\/p>\n
Think of another unbounded sequence: 1 million, 2 million, 3 million, …<\/p>\n
Is it possible for an infinite arithmetic sequence to be bounded? In other words, can you think of an example of a bounded arithmetic sequence? No. <\/p>\n
Reminder: an arithmetic sequence is a sequence whose terms grow by a fixed amount, the common difference. Here is an example of an arithmetic sequence whose common difference is 0.1: <\/p>\n
8, 8.1, 8.2, 8. 3, …<\/p>\n
For example, try to think of examples of: <\/p>\n
A sequence bounded between 0 and 1 that converges.
\n1\/5, 1\/25, 1\/125, …<\/p>\n
A sequence bounded between 0 and 1 that diverges.
\n0.5, 0.05, 0.95, 0.005, 0.995, 0.0005, 0.9995, …<\/p>\n
A geometric sequence that is bounded. Reminder: a geometric sequence has terms that are multiplied by a common ratio, for example: 2, 10, 50, 250, …
\n1, 1\/2, 1\/4, 1\/8, … is bounded between 0 and 2. <\/p>\n
A geometric sequence that is unbounded.
\n2, 10, 50, 250 above is unbounded. <\/p>\n
A geometric sequence that is bounded and converges.
\n1, 1\/2, 1\/4, … converges to 0. <\/p>\n
A geometric sequence that is bounded and diverges (does not converge).
\nNot possible. <\/p>\n
An arithmetic sequence that diverges.
\n1, 3, 5, 7, …<\/p>\n
An arithmetic sequence that converges.
\nNot possible. <\/p>\n
Remember sequence A? 16, 8, 4, 2, 1, 0.5, … Is sequence A bounded? If it is, what are three different examples of bounds for sequence A?
\nSequence A is bounded. Possible bounds: zero and 17, -1 and 16.1, zero and 20. <\/p>\n
The Fibonacci sequence is unbounded and diverges. <\/p>\n
The Rule Generated sequence is bounded and diverges. <\/p>\n
Theorem: Every unbounded sequence diverges. <\/p>\n
True. Think about how to explain why it is true. We will discuss explanations of this kind in future articles about logic and proof. <\/p>\n","protected":false},"excerpt":{"rendered":"
Consecutive Numbers Think of three consecutive whole numbers. 1, 2, 3 or 45, 46, 47 or 201, 201, 203 Think of three consecutive whole numbers bigger than a million: 2000000, 2000001, 2000002 Think of three consecutive odd numbers: 7, 9, 11 Think of three consecutive two-digit primes: 29, 31, 37 or 43, 47, 53 Think […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":159,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-168","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/168","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=168"}],"version-history":[{"count":1,"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/168\/revisions"}],"predecessor-version":[{"id":169,"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/168\/revisions\/169"}],"up":[{"embeddable":true,"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=\/wp\/v2\/pages\/159"}],"wp:attachment":[{"href":"https:\/\/bear-mathbelt.marjoriesayer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=168"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}