Scribe May 11/2007

Today, we were greeted, like everyday, with a POD! Today's problem was: a) Trace your direct ancestors back ten generations. How many people does that include? Ignore any inbreeding. b) How many generations must you go back to find 1 million ancestors.
Answers:
a) 2 ( 1 - 2^10/1-2) = 2046 We get this answer by using the geometric series formula, which in this class applies, because each generation we are multiplying the number of ancestors we have by two. When we divide the number of grandparents we have (4) by the number of parents we have (2), we get 2, so R = 2. N in this formula is equal to 10 because we are looking for how many ancestors we have ten generations back.
b) 1,000,000 = 2 [ 1-(2^n)/1-2]
500,000 = [1-(2^n)/1-2]
-500,000 = 1- (2^n)
-500,001 = (-2)^n
500,001 = 2^n Log2
500,001 = Log2 2^n
Log2 500,001 = n
Log 500,001/ Log 2 = 18.93 generations, or 19 generations.
In this part of the problem we are using a geometric series formula. Because we know Sn = 1,000,000, we can plug it in right away. Then, we figure out that T1 is equal to 2, because our first two direct ancestors are our parents. We know have two of the three variables that we can plug into this equation, so we are ready to solve for the third one, N. The algebra is pretty straight-forward, but, at the end of the problem we face a challenge, that most people in class were not able to figure out.Log2 500,001 = n... what do we do now?? Mr. Alcantara refreshed our memories, and showed us that:
if Log2 500,001 = y
then 2^y = 500,001
So, we solve now... Log 2^y = Log 500,001
y Log 2 = Log 500,001
y Log 2/Log 2 = Log 500,001/Log 2
y= Log 500,001/Log 2 y= 18.93
After explaining this to us, we agreed that there is no possible way to have 18.93 generations, you either have 18 or 19 generations, so, we decided, that to have one million direct ancestors, you must at least have 19 generations.
We then moved on into practice problems We had started to do these in class on Thursday, but we didn’t get to finish them, so we checked our answers today. The problem was: If T1 is = 1 and if T2 is = -3 Find S20
A) if the series is geometric b) if the series is arithmetic

a) The first thing you have to do is find R
r= T2/T1
r= -3/1
r= -3
Now, we can find S20.
S20 = 1 [1 – (-3)^20 / 1 – (-3)]
S20 = -871, 696, 100 .5

b) In b, the first thing you have to do is find d.
d = T2 –T1d= -3 – 1
d= -4
Now, we need to find T20, so that we can plug it into the arithmetic sequences equation.
T20 = 1 + (20-1) x -4
T20 = 1+ 19(-4)
T20 = 1 + -76
T20 = -75
Now, we have everything we need to plug into the arithmetic sequences equation.
S20 = (20/2) (1-75)
S20 = 10 x -74
S20 = -740

The second practice problem was: A grain of salt is placed on the corner square of a large chess board. Two grains are placed on the next square, four grains on the next, then eight, and so on. a) How many grains will the board have when half the squares have been covered? b) B) When all the squares have been covered? · A chessboard has 64 squares!

Answers:
a) S32 = 1 [1- (2^32) / 1-2]
S32 = 4,294,967,295 grains

We get his answer because r = 2 and we are looking for half the chess board, if it has 64 squares, then half the squares are 32 squares. We plug this information into a geometric sequences equation and we get the answer.

b) S64 = 1 [1- (2 ^64) / 1-2]
S64 = 1.8446744 X 10^19 grains.
We get this answer using the same information as in part a, only that we change 32 for 64 because we are looking for the total of the chessboard. After that, we were asked a new question using the same word problem…

c) How many grains were placed on the 16th square. We must realize that this is a geometric series question, it is not a geometric sequence, like the questions above.
T16 = 1 x 2^(16-1)
T16 = 1 (2^15) T16 = 2^15
T16 = 32, 768
We use the arithmetic series formula, because we only want the number of grains in square number 16, not the sum of the grains up to and including square 16.

The homework on the board is Pg 489 # 9-14, 17-20, 27, 29, 35 and, we must also post on the blog two word problems, one has to be an arithmetic series, and the other one a geometric series. Please post these problems without the solutions.
The next scribe will be… Chewy… who is alone… Have a nice weekend!! :D
Posted by: Cristina B and Pier