Scribe Post- Sofia and Juliana
03/05/07
On today’s class we started a new unit about series as a complement of the sequence unit, Rumidog started us off with a pod as usually.
POD
A business executive is offered one of the two raise schedules for the coming year.
Option 1: Receive a $100 raise each week
100,200,300
Option 2: Receive a #1000 raise each month
1000, 2000, 3000
Which option should she choose? How much will she make with that option compared to the other option?
At first we were really confused because we didn’t know any method that will have a short cut and be done as soon as possible. After several tries from all the students no one was able to create a short cut, but Martha came up with the answer after a long time with her calculator adding up. Later rumidog answer the POD pretty quickly.
Weekly: 100 * 26 * 53 = $137,800
Monthly: 1000 * 6 * 13 = $ 78,000
By subtracting $137,800 from $78,000it equals $59,800, so the best option was # 2
After the POD rumidog left us a bit confused but he assured us that later on we would understand how to produce short cuts for problems as the POD. So we moved on and started with a 2 minute problem. He asked us to sum all positive integers until 100. But obviously no one could solve the problem in the time he gave us. So he explained us better the purpose of the class by giving the following problem:
In your group make a sequence where the nth term is the sum of the first n positive integers. List the first 8 terms of the sequence.
1+2+3+4+5+6+7+8
The sequence was: 1, 3, 6, 10, 15, 21, 28, 36 we could observe a pattern so for the first term we add but later we just knew that the next term was going to be the same as the one before but we needed to add the next integer.
After this short problem we moved on and rumidog told us the story of Gauss and his abilities for math.
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1 + 2+ 3+ 4 ……………… 97+ 98+ 99+ 100
Pair up the terms from opposite end. 1 + 100 = 101 50 pairs and the sum is 101
Find the sum of each pair. 2 + 99 = 101 50(101) = 5,050
Multiply by the number of pair. 3 + 97 = 101
5,050 is the sum of all positive integers until 100
Gauss’s Formula:
1 +2 +3 …………….. n3 + n-2 + n-1 + n
Pair up the terms from opposite end.
Find the sum of each pair.
Multiply by the number of pair.
N + 1
2 + n -1 = n + 1
3 + n – 2= n + 1
n/2( n +1)
n/2 = #of pairs …… (n + 1) = sum of each pair
Arithmetic series: An indicated sum of the terms of an arithmetic sequence.
Sequence: 1,2,3,4,5
Explicit formula
Tn= n T4= 4
Series= 1+2+3+4
Explicit formula
Sn = n(n+1) / 2
S4= 4(5)/2
S4 = 10
Not all series are arithmetic.
Sn means “the sum of the first n term of the series” not just for arithmetic series
S5 for the Fibonacci series
1+1+2+3+5 S5 = 12
Find the S5 for the following arithmetic series:
5+10+15+20+25 S5=75
Gauss’s technique
5+10+15+20+25
5+25=30
10+20=30
5/2(30) = 75
HOMEWORK:
Pg. 489 # 1-8
3 comments:
sofi and juli, this is a very good post. I went to class that day, and I think you explained the material perfectly.
I was wondering if using the bubble sheet was easy. Did you have any difficulty?
Sofi and July,
What a great post. Indeed it helped me to recall what happened that day. I just arrived from a trip and can say that after I read you girls' scribe I am better than where i was. I know have a good start with Series. One final comment, besides what the evaluation Scribe Post says and what Rumidog may have recommended, I think that for future scribes, its better if the Scribe has colors, maybe to divide the POD with examples and Homework. What this does is that it makes it easier to understand visually.
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