MAT1830 : PRACTICE QUESTIONS ON INDUCTION
1.
Prove using induction that ∑𝑖=𝑛
𝑖=3 (2𝑖) = (n + 3)(n - 2) for all integers n > 3.
2.
Prove
by
induction
that
1  2 2  2  32  3  4 2  ...
to
n
terms
is
given
by
1
n n  1n  2 3n  5
12
3.
4.
4 n 2
 17 n  22 is divisible by 16 for every positive integer n.
Prove by induction 3
n
Prove that, if the statement n!  2n  2 is true for n = k, where k is a positive integer, then it is
true for n = k + 1.
Hence, find the set of values of n for which the statement is true.
5.
The sequence of real numbers u1 , u2 , u3 ,... is such that u1  1 and u n 1 
5u n  4
for all n  1 .
un  2
Prove by induction that un  4 for all n  1 .
6.
Let r0, r1, r2, …. be a recursive sequence defined by
r0 = 3; r1 = 2 and 𝑟𝑛 = (18n)𝑟𝑛−1 + (12)𝑟𝑛−2 for all integers n ≥ 2.
Prove using strong induction that 2𝑛 divides 𝑟𝑛 for all integers n ≥ 0.
7. If the sequence 𝑢1 ,𝑢2 ,𝑢3 , ….. is defined by 𝑢1 = 1, 𝑢2 = 2 and 𝑢𝑟+2 + 4𝑢𝑟 = 4𝑢𝑟+1 ,
Prove by induction that 𝑢𝑛 = 2𝑛−1
8. Prove by induction that the number of steps to complete the disk movements in the tower ofHanoi
problem is ( 2𝑛 – 1) for a system of n disks, for all n≥1