09-10 Questions

[thatoneguy]

Which of the following is both a happy and a perfect number, 7, 28, or 42?

The set {l,u,c,a,s} has ___ proper subsets?

The simplified coefficient of the xy^2 term in the expansion of (2x + 3y0^3 is?

(67 base 9 + 84 base 9)/8 =?

The sum of the first 9 terms of the Fibonacci characteristic sequence 1,4,5,9,14,23 is?

Thanks guys

[nsguy1350]

You should know that 6,28,496,and 8128 are perfect, so you don't have to worry about the 'happy' part, because 28 is the only perfect number. Some people memorize happy numbers, but I have neglected to do so. I might in the future.

Proper subsets = 2^(number of elements) - 1 = 2^5 - 1 = 31.
I plug the coefficents of x and y into their respective powers, then multiply by the binomial coefficient. You can memorize the binomial coefficients to a point, or do combinations, but memorizing helps a lot. You answer is (2)(3)^2 (binomial coefficient = 3) = 54. For (x+y)^3, the coefficients of the expansion are 1,3,3,1, just in case you didn't understand what I meant by binomial coefficient. Hopefully you recognize those numbers.

I would just convert to base ten first, unless the answer is in base 9, unless you'd like to add the numbers first. This becomes 162 base 9, or 137 base 10 divided by 8, which is 17 1/8.

The sum of the first nine terms of a Fibonacci characteristic sequence [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] represents the nth Fibonacci number, with [unparseable or potentially dangerous latex formula] etc, is [unparseable or potentially dangerous latex formula].

The ninth Fibonacci is 34, while the tenth is 55. The solution is: (34)(1) + (55-1)(4) = 34 + 216 = 250.