Download Exercises: Use Induction. 1). Show that the sum of the

Exercises: Use Induction.
1). Show that the sum of the first n consecutive even natural numbers is n(n + 1). (Hint:
Begin by finding a formula for the nth even natural number.)
2). Guess a formula for the sum of the first n consecutive odd natural numbers. Use
induction to prove that your formula is correct.
3) Show 1 + 4 + 9 + 16 + · · · + n2 = n(n + 1)(2n + 1)/6 for all natural numbers n.
4) Show 1 + 8 + 27 + 64 + · · · + n3 = n2 (n + 1)2 /4 for all natural numbers n.
5) Show 1 + 2 + 4 + 8 + 16 + · · · + 2n−1 = 2n − 1 for all natural numbers n.
6) Show 1 + 3 + 9 + · · · + 3n−1 = (3n − 1)/2 for all natural numbers n.
7) Show 2n > n for all natural numbers n.
8) Show n! > 10n for all natural numbers n ≥ 5. Recall that n! is defined to be the product
of the numbers from 1 to n. So 2! = 2 · 1 and 3! = 3 · 2 · 1.
9) The Tower of Hanoi is a popular game consisting of n discs with decreasing diameters
piled up on a spindle. There are two empty spindles nearby. The object is to move the
discs one by one from the given spindle to one of the adjacent spindles, never placing a
larger disc on top of a smaller one. Show that one can move the discs in 2n − 1 moves.
10) Show
1
1·2
11) Show
1
1·2·3
+
1
2·3
+
+
1
3·4
+ ··· +
1
2·3·4
+
1
3·4·5
1
n·(n+1)
+ ··· +
=
n
n+1
for all natural numbers n.
1
n(n+1)(n+2)
=
n(n+3)
4(n+1)(n+2)
for all natural numbers n.
12). Students are waiting in line to buy tickets to a rock concert. The first student in line
is a woman and the last student is a man. Prove that somewhere in the line there is a
man standing directly behind a woman. (Hint: let n be the number of students, and use
induction starting with n = 2.)
13) Consider a chess board that has size 2n × 2n . Draw a picture of this board for n =
1, 2, 3, 4, and imagine what this board looks like for larger n. Somebody has painted one
of the squares on the board green. Show that you can completely cover the remaining
squares, leaving only the green square exposed, using non-overlapping L-shaped tiles, each
L-tile large enough to cover exactly 3 squares.
14) Let θ be an angle. Show that for every natural number n
sin(θ) = 2n · sin(
θ
θ
θ
θ
θ
)
·
cos(
)
·
cos(
)
·
cos(
)
·
·
·
cos(
)
2n
2
4
8
2n