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Monday, June 05, 2006

SCRIBE: Sequences & Series

Well Michael ended off his scribe with a coded sequence to reveal who the next scribe with so I'll show you how we found out it was me. 14, 34, 39, 43, 47, 56, 61.

The differences between each numbers are as follows

20, 5, 4, 4, 9, 5

Since he told us that the differences represented the order of the alphabets, we get this word.

T E D D I E

So today's scribe is me Teddie!

Okay enough with the fun and game and lets get down to the stuff we did today cuz it was a whole lot.


THE MATH DICTIONARY

SEQUENCES & SERIES

Sequence: An ordered list of numbers that follows a certain rule or pattern.

Series: The sum of numbers in a sequence to a particular term.

Example: Given the sequence
3, 6, 12, 24, 48, . . .

S1 = 3
S
2 = 3 + 6 = 9
S3 = 3 + 6 + 12 = 21
S4 = 3 + 6 + 12 + 24 =
S5 = 3 + 6 + 12 + 24 + 48 = Sn = 3 + 6 + 12 + 24 + 48 + . . . . tn
S5 Denotes the sum of the first 5 terms.
S
n denotes the sum of the first n terms.

ARITHMETIC SEQUENCE

I )
Recursive Definition: an ordered list of numbers generated by continually adding a value (the
common difference) to a given first term.

II ) Implicit Definition: an ordered list of numbers where each number in the list is generated by a linear equation.

III ) Common difference (d):
a)
the number that is repeatedly added to successive terms in an arithmetic sequence.
b) from the implicit definition d is the slope of the linear equation.

To Find a Common Difference

d = tn -t(n-1)

d is the common difference
t
n is any term in the sequence other than the first term.
t1 is the first term in the sequence

Finding the nthe Term in an Arithmetic Sequence

tn = t1 + (n - 1)d

tn is the nthe term in the sequence
t1 is the first term n is the rank of the term
d is the common difference

Example: find the 51
>st term of the arithmetic sequence 11, 5, -1, -7, . . . .

Solution:

n = 51
t1 = 11
d = -6

t51 = t1 + (n - 1)d
= 11 + (51 - 1)(-6)
= 11 + (50)(-6)
= 11 - 300
= -289

ARITHMETIC MEANS

The term(s) that falls between any two non-consecutive terms in an arithmetic sequence

Examples: Find the arithmetic means in each sequence
a) 1, _ , 25
there are 2 gaps for differences to occur
25 - 1 = 24
2d = 24
d = 12

1, 13 , 25

b) 3, _ , _ , _ , 1
there are 4 gaps for differences to occur
1 - 3 = -2
4d = -2
d = -0.5

3, 2.5, 2, 1.5, 1

Arithmetic Series: the sum of numbers in an arithmetic sequence given by:

Sn = (n/2) [ 2t1 + (n - 1) d ]

d is the common difference
S
n is the sum to the nthe term
t
1 is the first term

Example:
find the sum to the 21
st term of the sequence
5, 9, 13, 17, . . .

Solution:
t1 = 5
d = 4
n = 21

Sn = (n/2) [ 2t1 + (n - 1) d ]
S21 = (21/2) [ 2(5)
+ (21 - 1)(4) ]
S21 = (21/2) [ 10 + 80 ]
S21 = (21/2) [ 90 ]
S21 = 21 [ 45 ]
S21 = 945

GEOMETRIC SEQUENCES:

I ) Recursive:
an ordered list of numbers generated by repeatedly multiplying a given first term by a particular value (the common ratio).

II ) Implicit:
the ordered list of numbers where each number in the list is generated by an exponential function

And that is all for the notes in our dictionary, pew.

Starting off the afternoon we started off with some problems.

1. Find Sn for each arithmetic sequence

a) t
1 = 8
d = 5
S
25 = ?

S25 = (25/2) [ 2(8) + (25 - 1) 5 ]
S25 = (25/2) [ 16 + 120 ]
S25 = (25/2) [ 136 ]
S25 = (25) [68]
S25 = 1700

b) t
5 = 15
t
19 = 43
S
100 = ?
d = ?
t
1 = ?

19 - 5 = 14 common differences required
43 - 15 = 28
14d = 28
d = 2

t
5 = t1 + (5 - 1) d
15 = t
1 + (4)(2)
15 = t
1 + 8
7 = t
1

S
100 = (100/2) [ 2(7) + (100 - 1)(2) ]
S100 = 50 [ 14 + 198 ]
S100 = = 50 [212]
S100 = 10600

2. Given
S12 = 360, t1 = -3 find d
S12 = (12/2) [ 2 t1 + (12 - 1) d ]
360 = 6 [ 2(-3) + 11d ]
360 = -36 + 66d
396 = 66d
d = 6

3. Given S
30 = 480 t1 = 4 find t30

S30 = (30/2) [ 2t1 + (30 - 1) d ]
480 = 15 [ 2(4) + 29d ]
480 = 120 + 435d
360 = 435d
24/29 = d

t
30 = t1 + (30 - 1) d
t
30 = 4 + 29(24/29)
t
30 = 4 + 24
t
30 = 28

4. x = 1, (1/2)x + 4, 1 - 2x form and arithmetic sequence what are the first 3 terms?

[(1/2)x + 4] - (x - 1) = (1 - 2x) - [(1/2)x + 4]
1/2)x + 4 - x - 1 = 1 - 2x - (1/2)x + 4
2x = -8
x = -4
x - 1 = -4 -1 = 5
(1/2)x + 4 = (1/2)(-4) + 4 = 2 1 - 2x = 1 - 2(-4) = 9

Finally I am done yay! And tomorrow's scribe is CALVIN!





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1 Comments:

At 6/06/2006 8:35 PM, Blogger Mr. H said...

Nice Scribe. I would consider it for the Hall of fame if you were consistant with your colour choices one for definition and one for examples and if you added links to other sites where this information can be found.

The layout of the scribe is excellent just work on the colour choices and try to spice it up with a link or two!!

Email me when you have completed your mini task to be in the Hall.

charbeck@wsd1.org
Mr. Harbeck
Sargent Park School

 

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