Math Ia - Infinite

Cynthia Li
IB Mathematics SL 2
Mr. Panych
September 8, 2009

Internal Assessment: numberless Surds

Infinite surds give be used in this investigation to stipulate a general statement for all integer set of the infinite surd .
First, a rule was found for a1= , a2= , a3= . From this sequence, a general rule for an+1 is an+1= . The value of the premier(prenominal) ten terms of this sequence were calculated.
nValue

11.4142
21.5538
31.5980
41.6118
51.6161
61.6161
71.6174
81.6179
91.6180
101.6180







It under side of meat be seen from the data to a higher place that as the value of n increases, the value of the sequence increases, solely at a decaying rate. This observation is solidified upon graphing the relationship amidst n and an. The graph crowd out be found on the next scalawag.






The relationship noted on the previous page is confirmed with this graph. After the infinite surd reaches its fourth term, the passing between each term and the previous one decreases. an is approach path a constant value, which is the critical value of the sequence. This also suggests that an an-1 will reach zero eventually, when an is large enough.
Next, we must find the exact value of this surd. First, we make the infinite surd equal to X.




X =
When the entire equation is squared, we have

Since we know X = , we can derive the equation

Subtracting 1+x from each side of the equation leaves us with the quadratic .


The quadratic equation , can now be used to solve this equation. Substituting the values 1, -1, and -1 for a, b, and c respectively, we get the equation
, which simplifies to

However, the exact value of this surd must be positive, because the graph is approaching a positive number. Therefore, we teach the positive solution of , which is .

Now that we have found the number for , we follow the same steps to test our findings for . First, we find the quantitative values for the first ten terms of this sequence.

nValue
11.8477
21.9620...If you want to get a full essay, enunciate it on our website: Ordercustompaper.com



If you want to get a full essay, wisit our page: write my paper