Expert: Sherry Wallin - 1/2/2010

Question
WHAT ARE INFINITE SURDS?!
I am asked the following question:
a1=sqrt(1 + sqrt 1)
a2=sqrt (1+ sqrt 1 + sqrt 1)
a3=sqrt (1+ sqrt 1 + sqrt 1 + sqrt 1)
etc.
Find a formula for an+1 in terms of an. <--- (HOW AM I SUPPOSED TO DO ATHAT?)
Calculate the decimal values of the first ten terms of the sequence. Using technology plot the relation between n and an. Describe what you notice. What does this suggest about the value of an-an-1 as n gets very large? Use your results to find the exact value for this infinite surd. (WHAT IS EXACT VALUE?- HOW DO I FIND EXACT VALUE)


now consider the general infinite surd sqrt(k + sqrt k + sqrt k) where the first term is sqrt (k + sqrt k). Find an expression for the exact value of this general infinite surd in terms of k. (How DO I FIND EXACT VALUE?!)

Find some values of k that make the expression an integer. find the general statement that represents all the values of k for which the expression is an integer. (HOW DO I FIND THE GENERAL STATEMENT?!)
Test the validity of your general statement using other values of k.
discuss the scope and or limitations of your general statement.
Explain how you arrived at your general statement.

ANY HELP is appreciated!
thankss!

THATS MORE SPECIFIC!

Answer
Ca you calculate the first 10 terms of either? I think you have it written wrong by the way:
Your a1 is correct but a2=sqrt(1 + sqrt 1)It is sqrt(1 + sqrt(1+sqrt (1))) etc

Math Prof

The exact value is what the series will converge to when you add a sufficient amount. It really isn't exact, but you can find the answer to the nearest .001 or .0001 or .00001 or whatever. You will never get the exact.