Download MATH 1200 The list of sample questions for test 1 in winter semester

MATH 1200
The list of sample questions for test 1 in winter semester.
1. Prove that for all n ≥ 1, 1 + 4 + . . . + (3n − 2) =
n(3n−1)
.
2
2. Prove that for all positive integers n, 7n − 2n is divisible by 5.
3. Prove that for any positive integer n, 2(2) + 3(22 ) + . . . + (n + 1)(2n ) = n(2n+1 ).
4. Prove that for every non-negative integer n, 2n > n.
5. Show that for any positive integer n, 2n3 + n is divisible by 3.
6. Show that for any positive integer n, 6n − 1 is divisible by 5.
7. Prove that
Pn
1
i=1 i(i+1)
=
n
n+1 .
8. Prove that for any real number X > −1, and any positive integer n, (1 + X)n ≥ 1 + nX.
9. Show that the sum of first n positive integers is
n(n+1)
.
2
10. Prove that for any positive integer n, 13 + 23 + . . . + n3 =
n2 (n+1)2
.
4
11. (a) Prove by induction that if b is an odd number and n a positive integer, then bn is also odd.
(b) Using the conclusion of the first part of this question, show that the equation x19 + x + 1 = 0 has no
solutions that are integers.
12. Show that there are non-negative integers a and b such that 3a + 2b = n for any integer n ≥ 5.
13. Prove that every positive integer n > 1 is product of some prime numbers.
14. Prove that for every non-negative integer n, 12|(n4 − n2 ).
15. Let F1 = 1, F2 = 1 and Fn = Fn−1 + Fn−2 (this sequence is the well-known Fibonacci sequence). Show
that for every positive integer n, Fn < 2n .
1