MTH 311, Problems Type II, set 1, discussed 10/3
1. Exercises from the textbook: Note: Numbers like 1.1/7/p.11 refer to the
exercise #7 listed after Section 1.1 of the textbook on p.11.
•1.1/2,10/p.12
•1.2/6/p.17.
•1.3/1,9/p.23
•1.4/1,9/p.32.
2. Show by a direct method:
A: For all integers kand l, if k, l are both even, then kl is even.
Is the converse true ? If yes, prove it. If not, try to formulate a correct
statement which will use “if kl is even” as its hypothesis. Can you weaken
the hypothesis of A ?
3. Prove by contradiction:
B: For all integers kand l, if kl is odd, then k, l are both odd.
4. Try proving A by contradiction and B by a direct method.
5. Fibonacci numbers {Fn}nare defined as follows F0= 0, F1= 1 and
Fn=Fn−1+Fn−2, for n≥2. Show using mathematical induction that
Pn
i=1 F2
i=FnFn+1, for n≥1.
6. Harmonic numbers {Hn}nare defined as follows Hn= 1 + 1/2 + · · · 1/n.
Show that 1 + n/2≤H2n≤1 + n.
7. For what nis it true that 2n< n! ? Replace 2 by kand answer the same
question. Prove a general statement by induction.
8. Prove that for any n≥14, ncan be written as a sum of 3’s and/or of 8’s
(with no regard to order).
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