Summer 2008 CMSC 351: Homework 0 Clyde Kruskal
Problem 1. Use mathematical induction to show that
(a)
n
X
i=1
i(i+ 1) = n(n+ 1)(n+ 2)
3(b)
n
X
i=0
2i= 2n+1 −1
Problem 2. See bottom of page 53 of CLRS (bottom of page 34 in CLR) and/or the
bottom of this sheet.
(a) Assume bx=a. What is x(in terms of aand b)?
(b) Using only part (a), show that logc(ab) = logca+ logcb.
(c) Show that alogbn=nlogba
Problem 3. Differentiate the following functions:
(a) ln(x2+ 5)
(b) lg(x2+ 5)
(c) 1
ln(x2+5)
Problem 4. Integrate the following functions:
(a) 1
x
(b) 1
3x+7
(c) ln x[HINT: Use integration by parts.]
(d) xln x[HINT: Use integration by parts.]
(e) xlg x
lg n= log2n
ln n= logen
lgkn= (lg n)k
lg lg n= lg(lg n)
For all real a > 0, b > 0, c > 0, and n,
a=blogba
logc(ab) = logca+ logcb
logban=nlogba
logba=logca
logcb
logb(1/a) = −logba
logba=1
logab
alogbn=nlogba