2.5 The Twin Paradox and the Good Old Logarithm 41Review Exercises
2.5.1What is the following sum?
1
1 · 2
+
1
2 · 3
+
1
3 · 4
+···+
1
(n−1)·n
.Experiment, conjecture the value, and then prove it by induction.
2.5.2What is the following sum?0 ·
n
0
+1·
n
1
+2·
n
2
+···+(n−1)·
n
n− 1
+n·
n
n
.Experiment, conjecture the value, and then prove it. (Try to prove the result by
induction and also by combinatorial arguments.)
2.5.3Prove the following identities:1 · 20 +2· 21 +3· 22 +···+n· 2 n−^1 =(n−1)2n+1.13 +2^3 +3^3 +···+n^3 =
n^2 (n+1)^2
4
.1+3+9+27+···+3n−^1 =
3 n− 1
2
.2.5.4Prove by induction onnthat
(a)n^2 −1 is a multiple of 4 ifnis odd,
(b) n^3 −nis a multiple of 6 for everyn.2.5.5There is a class of 40 girls. There are 18 girls who like to play chess, and
23 who like to play soccer. Several of them like biking. The number of those who
like to play both chess and soccer is 9. There are 7 girls who like chess and biking,
and 12 who like soccer and biking. There are 4 girls who like all three activities.
In addition we know that everybody likes at least one of these activities. How
many girls like biking?
2.5.6There is a class of all boys. We know that there areaboys who like to
play chess,bwho like to play soccer,cwho like biking anddwho like hiking. The
number of those who like to play both chess and soccer isx. There areyboys
who like chess and biking,zboys who like chess and hiking,uwho like soccer
and biking,vboys who like soccer and hiking, and finallywboys who like biking
and hiking. We don’t know how many boys like, e.g.,chess, soccer and hiking, but
we know that everybody likes at least one of these activities. We would like to
know how many boys are in the class.
(a) Show by an example that this is not determined by what we know.
(b) Prove that we can at least conclude that the number of boys in the class
is at mosta+b+c+d, and at leasta+b+c+d−x−y−z−u−v−w.