Monday, June 2, 2008

Geometric Sequences

WARNING: The geometric sequences unit does not have any new material; rather the unit gives a new perspective to what we already know from previous learning.

OVERVIEW:
• What is a sequence?
• Recursive definition vs. implicit definition
• Common difference vs. common ratio
• Determining any term (the nth term) in an arithmetic sequence vs. determining any term (the nth term) in a geometric sequence

SLIDE 2 An example of an arithmetic sequence is the first sequence of numbers found on SLIDE 2. Examples of geometric sequences is the second and the third sequence of numbers. Note that arithmetic sequences and geometric sequences aren't the only sequences that exists, such as the Fibonacci sequence, but the main scope of the unit focuses on geometric sequences.

We find the next three terms by determining a rule for each sequence of numbers.
1. For the first sequence of numbers (4, 7, 10, 13...), we see that 3 is added to any term to get the next term.
2. For the second sequence of numbers (3, 6, 12, 24...), we see that 2 is multiplied to any term to get the next term.
3. For the third sequence of numbers (32, 16, 8, 4...), we see that 1/2 is multiplied to any term to get the next term. (Remember, in grade 12, we multiply by 1/2 instead of dividing by 2; it's to make our calculations easier to do.)
4. For the fourth sequence of numbers (1, 1, 2, 3...), we see that we have to add the previous two terms to get the next term.

SLIDE 3 Let's take a closer look at the first sequence of numbers (4, 7, 10, 13...).

How did we determine that the rule is to add 3 to any term to get the next term? We find that out by determining that the common difference is 3 (green font). We then find out that if we're given any term, n, then 3n+1 is the equation we use to get what n equals. For example,

3n+1 = 3(1)+1 = 4.

So if the rank (n) is 1, then its term is 4. If the rank is 2, then its term is 7. If the rank is 7, then its term is 22. This is expressed with the equation 3n+1.

The rank represents the term. For example, in this sequence, the rank of the first term is 1, the rank of the second term is 2, the rank of the third term is 3, the rank of the nth term is 3n+1, and so on.

Graph 3n+1, and we find that it is the equation of a line. Also, we can determine the 0th term by plugging 0 into the equation.

3n+1 = 3(0)+1 = 1.

For the zeroth term, the output 1. It is also the y-intercept of the graph.

Term 1 (t1), or in some textbooks it's called a, equals 4. The common difference, d, is 3.

The recursive definition, in this sequence of numbers, is to take a term and add 3 continuously.

The implicit definition, in this sequence of numbers, is the equation of the sequence, tn = 3n+1.

SLIDE 4 Let's take a closer look at the second sequence of numbers (3, 6, 12, 24...).

How did we determine that the rule is to multiply 2 to any term to get the next term? Firstly, there isn't a common difference in this case but a common ratio. The common ratio is 2. We then find out that if we're given any term, n, then (3/2)(2^n) is the equation we use to get what n equals.

tn = (3/2)(2^n) = [3 * (1/2) * 2^n] = [3 * 2^(n-1)]

(Refer to the bottom-right corner of SLIDE 4.)

So if the rank (n) is 1, then its term is 3. Similarly, if the rank is 2, then its term is 6. If the rank is 7, then its term is 192. Etcetera.

Graph [3 * 2^(n-1)], and we find that it is the equation of an exponential function. Also, we can determine the 0th term by plugging 0 into the equation.

tn = [3 * 2^(n-1)]
t(0) = [3 * 2^(0-1] = 3/2

For the zeroth term, the output 3/2. It is also the y-intercept of the graph.

The recursive definition, in this sequence of numbers, is to take a term and multiply it continuously by 2.

The implicit definition, in this sequence of numbers, is the equation of the sequence, tn = 3*2^(n-1).

SLIDE 5 Let's take a closer look at the third sequence of numbers (32, 16, 8, 4...).

The recursive definition, in this sequence of numbers, is to take a term and multiply it continuously by 1/2.

The implicit definition, in this sequence of numbers, is the equation of the sequence.

VOCABULARY/SUMMARY is found on SLIDES 6 to 9.

HOMEWORK
• Exercise 45: Geometric Sequences

Next Scribe is Nelsa.