7/31/2019 A Sequence is a Function Whose Domain is the Natural
Numbers
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Fibonacci Sequence
One famous example of a recursively defined sequence is the
Fibonacci Sequence. The first two terms ohe Fibonacci Sequence are
1 by definition. Every term after that is the sum of the two
preceding terms.The Fibonacci Sequence is 1, 1, 2, 3, 5, 8, 13, 21,
34, 55, 89, ... a n+1 = an + an-1.Fibonacci sequences occur
frequently in nature. For example, take a leaf on a stem of many
plants (likecherry, elm, or pear trees). Count the number of leaves
until you reach one directly in line with the one yoselected. The
total number of leaves (not including the first one) is usually a
Fibonacci number. If the left ight handed spirals on a pine cone,
sunflower seed heads, or pineapples are counted, the two numbers
a
often consecutive Fibonacci numbers.
Factorials - !
The symbol for factorial is ! (an explanation point). The
factorial of a positive integer is the product of allpositive
integers less than or equal to that number. Zero factorial is a
special case and 0! = 1 by defintion.! = 1
2! = 2*1! = 2*1 = 23! = 3*2! = 3*2*1 = 64! = 4*3! = 4*3*2*1 =
245! = 5*4! = 5*4*3*2*1 = 120n! = n(n-1)! =
n*(n-1)*(n-2)*...3*2*1
Notice that there is also a recursive definition in there. Any
number factorial is that number times the factoof one less than
that number.On the TI-82 and TI-83 calculators, the factorial key
can be found under Math, Probability, menu choice 4
Simplifying ratios of factorials
Consider 8!/5!. One way to work the problem would be to fully
expand the 8! and fully expand the 5!.8! 8 * 7 * 6 * 5 * 4 * 3 * 2
* 1- = ----------------------------- = 8 * 7 * 6 = 336
5! 5 * 4 * 3 * 2 * 1Then you notice that there are common
factors of 5, 4, 3, 2, and 1 in both the numerator and
denominator
hat divide out. This leaves you with 8*7*6 in the numerator for
an answer of 336.Another way to work the problem, however is to use
the recursive nature of factorials. Since any numberactorial is
that number times the factorial of one less than that number, 8! =
8 * 7!, but 7! = 7 * 6!, and 6! =5!. This means that 8! = 8 * 7 * 6
* 5!. So, 8! / 5! Is 8*7*6*5!/5! = 8*7*6 = 336.
The point is that there is no need to multiply the entire thing
out when you're just going to be dividing abunch of it out
anyway.Here are some steps to simplifying the ratio of two
factorials.
1. Find the smaller factorial and write it down. It won't
change, and it will be the part that is completelydivided out.
2. Take the larger factorial and start expanding it by
subtracting one until the smaller number (that
you've already written down) is reached.3. When you reach the
smaller number, write it as a factorial and divide out the two
equal factorials.
Here's an example with a little bit more complicated ratio.
Since 2n-1 is less than 2n+1, copy the 2n-1actorial down on the
bottom. Now, take 2n+1 times one less than 2n+1. One less then 2n+1
is 2n+1-1=2nTake that times one less than 2n, which is 2n-1. Since
that 2n-1 is the same amount on the bottom, call th2n-1 factorial
and then divide it out, leaving the product of 2n and 2n+1 as the
answer.2n+1)! (2n+1)*(2n)*(2n-1)!------ = ------------------- =
2n(2n+1)2n-1)! (2n-1)!
7/31/2019 A Sequence is a Function Whose Domain is the Natural
Numbers
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Sigma / Summation Notation
Summation is something that is done quite often in mathematics,
and there is a symbol that meanssummation. That symbol is the
capital Greek letter sigma, and so the notation is sometimes called
SigmaNotation instead of Summation Notation.
The k is called the index of summation. k=1 is the lower limit
of the summation and k=n (although the k isonly written once) is
the upper limit of the summation.
What the summation notation means is to evaluate the argument of
the summation for every value of thendex between the lower limit
and upper limit (inclusively) and then add the results
together.
Examples:
Substitute each value of k between 1 and 5 into the expression
3k-2 and then add the results together.3(1)-2] + [3(2)-2] +
[3(3)-2] + [3(4)-2] + [3(5)-2] = 1 + 4 + 7 + 10 + 13 = 35
3(1) + 3(2) + 3(3) + 3(4) + 3(5) ] - 2 = [ 3 + 6 + 9 + 12 + 15 ]
- 2 = 45 - 2 = 43Although these look very similar, the answers are
different. The second example is thrown in there to waryou about
notation. Multiplication has a higher order of operations than
addition or subtraction, so no grousymbols are needed around the
3k. But since subtraction has the same precedence as addition,
thesubtraction of 2 does not go inside the summation. In other
words, be sure to include parentheses aroundsum or difference if
you want the summation to apply to more than just the first
term.
Properties of Summation
The following properties of summation apply no matter what the
lower and upper limits are for the index. simplicity sake, I will
not write the k=1 and n, but know that your index of summation is k
in the followingexamples.
You can factor a constant out of a sum.
cak = cakNotice the ak has a subscript of k while the c doesn't.
This means that the c is a constant and the a isunction of k. The
sum of a constant times a function is the constant times the sum of
the function.
The sum of a sum is the sum of the sums
Ooh, that just sounds good. The summation symbol can be
distributed over addition.(ak + bk) = ak + bk
The sum of a difference is the difference of the sums
The summation symbol can be distributed over subtraction.(ak -
bk) = ak - bk
