Math 310/520
2–27–2009
Problem Set 10
•This is a writing assignment. You will be graded on mathematical correctness, but also on your
writing. In order to receive full credit for the writing, your entire solution must be typeset —
not handwritten. When you get your work back, you will be allowed to revise it and resubmit your
revised work to improve your score.
•Problems marked “[MATH 520]” are to be done by Math 520 students; Math 310 students will not get
credit for doing them. Other problems are to be done by everyone.
[Problems on Induction]
1. A sequence of integers is defined by
x0= 9, x1= 3,
xn= 3xn−1+ 10xn−2for all n≥2.
Prove that
xn= 3 ·5n+ 6 ·(−2)nfor all n≥0.
2. Use induction to prove that
5n≥4n+ 2 for all n≥2.
•There are various ways to do this. Remember the obvious fact that 5 >4.
[MATH 520] 3. Prove that if nis an integer and n≥1, then
1·1! + 2 ·2! + ···+n·n! = (n+ 1)! −1.
Don’tstartwithwhatyou’re
tryingtoprove.
A common symptom of this is a proof that ends in “0 = 0”, “1 = 1”, “x2=x2”, ....You will lose
half the credit for a problem if you start with what you want to show.
•In class, we discussed a writing error which involves continuing a series of equations after stating the
n+ 1 statement (the “big period”). If you make this error, you’ll lose 0.5 points per problem.
The reward of a thing well done is to have done it. -Ralph Waldo Emerson
c
2009 by Bruce Ikenaga 1