What Is the Digit to the Right of the Decimal Point in (sqrt(3) + sqrt(2))^2002?

[gfleming]

Homework Statement

The following is a question from a set-text that I have chosen to explore.

What digit is imediately to the right of the decimal point in (sqrt(3) + sqrt(2))^2002

Homework Equations

The Attempt at a Solution

I have not gone very far with this, and may need to find a different problem. What I have done is explore (sqrt(3) + sqrt(2))^n, for n = 0, 1, 2, 3, 4... and found that for even values of n (which include 2002) that the expression evaluates to the sum of a constant and a constant multiple of sqrt(6). Both constants are too large to calculate.

I also know that [(sqrt(3) + sqrt(2))^2002][(sqrt(3) - sqrt(2))^2002]=1, but I am not sure if that can be used to find a solution.

I am presently exploring [(sqrt(3) + sqrt(2))^2002]-[(sqrt(3) - sqrt(2))^2002] and find that a whole lot of terms are being cancelled.

Any suggestions would be appreciated,

Thanks.

 

[mighty2000]

You are on the right track with your attempts so far. It is important to note that for any even value of n, the expression (sqrt(3) + sqrt(2))^n will always evaluate to a sum of a constant and a constant multiple of sqrt(6). This is because when you expand the expression, you will always get a combination of terms that involve both sqrt(3) and sqrt(2) raised to even powers, which will simplify to a constant. The remaining terms will involve both sqrt(3) and sqrt(2) raised to odd powers, which will simplify to a constant multiple of sqrt(6).

You are correct in exploring the expression [(sqrt(3) + sqrt(2))^2002]-[(sqrt(3) - sqrt(2))^2002], as this will cancel out all the terms involving constants and leave you with just the terms involving sqrt(6). However, this may not be the most efficient way to find the digit immediately to the right of the decimal point. One approach you can try is to use the binomial theorem to expand the expression (sqrt(3) + sqrt(2))^2002 and then look for patterns in the coefficients of the terms involving sqrt(6). This may lead you to a more direct way of finding the desired digit.

I hope this helps and good luck with your exploration!


Scientist