PRACTICE QUESTIONS ON MATHEMATICAL INDUCTION
1) Show that
2) Show that
3) Show that
n
X
k3 =
k=1
n
X
k(k + 1) =
n(n + 1)(n + 2)
for all n ∈ Z+ .
3
(−1)k k 2 =
(−1)n n(n + 1)
for all n ∈ Z+ .
2
k=1
n
X
k=1
4)
5)
6)
7)
8)
n(n + 1) 2
for all n ∈ Z+ .
2
Prove that n3 ≤ 2n for all n ∈ Z+ with n ≥ 10.
Prove that 11n − 6 is divisible by 5 for all n ∈ Z+ .
Show that n3 − n is divisible by 6 for all n ∈ Z+ .
Show that n5 − n is divisible by 5 for each n ∈ Z+ .
(Slightly trickier) Prove the binomial theorem: that is, show that
n X
n k n−k
(a + b)n =
a b
k
k=0
holds for all a, b ∈ R and n ∈ Z+ .
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: [email protected]
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