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 of three consecutive perfect squares: 16, 25, 36 or 49, 64, 81

Patterns

A. 1, 10, 2, 20, 3, 30, 4, 40, 5, 50, …

B. 1, 10, 19, 28, 37, 46, 55, 64, 73, 82, …

C. 1, 10, 100, 1000, 10000, 100000, …

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, …

(This can be written as 1, 1/10, 1/100, 1/1000, …)

What is another geometric sequence where the terms get smaller and smaller? Think of one.

81, 9, 1, …

What day of the week will it be in 1473 days? (If it is Wednesday today): Saturday

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.

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.

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.

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.

Rule 4: If the number is greater than 10, subtract 6 to get the next term.

If the first term is 9, what is the 100th term? 9

If the first term is 2, what is the 100th term? 11

Destinations

Can you think of another sequence that goes to zero?
0.2, 0.02, 0.002, 0.0002, …

or

1, -1, 1/2, -1/2, 1/3, -1/3, …

Can you think of another sequence that goes to infinity? 1, 3, 9, 27, …

or

2, 3, 5, 7, 11, 13, …

1, 1.01, 1.02, …

Think of an infinite sequence that converges to 10: 9, 9.9, 9.99, 9.999, …

or

10.1, 10.01, 10.001, …

Think of an infinite sequence that converges to 10 in an unusual way:

10 + 1/2, 10 + 1/3, 10 + 1/5, 10 + 1/7, 10 + 1/11, 10 + 1/13, …

Think of another unbounded sequence: 1 million, 2 million, 3 million, …

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.

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:

8, 8.1, 8.2, 8. 3, …

For example, try to think of examples of:

A sequence bounded between 0 and 1 that converges.
1/5, 1/25, 1/125, …

A sequence bounded between 0 and 1 that diverges.
0.5, 0.05, 0.95, 0.005, 0.995, 0.0005, 0.9995, …

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, …
1, 1/2, 1/4, 1/8, … is bounded between 0 and 2.

A geometric sequence that is unbounded.
2, 10, 50, 250 above is unbounded.

A geometric sequence that is bounded and converges.
1, 1/2, 1/4, … converges to 0.

A geometric sequence that is bounded and diverges (does not converge).
Not possible.

An arithmetic sequence that diverges.
1, 3, 5, 7, …

An arithmetic sequence that converges.
Not possible.

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?
Sequence A is bounded. Possible bounds: zero and 17, -1 and 16.1, zero and 20.

The Fibonacci sequence is unbounded and diverges.

The Rule Generated sequence is bounded and diverges.

Theorem: Every unbounded sequence diverges.

True. Think about how to explain why it is true. We will discuss explanations of this kind in future articles about logic and proof.