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W.1 PURE MATHEMATICS (MATHEMATICAL INDUCTION)
Ex.A
(n3 + 20n) is divisible by 48 for all positive even integers n.
1.
Prove, by induction, that
2.
For any positive integer n , let

bn = 1 + 3
n + 1 - 3 n
an =

 

n
n
1 
1+ 3 - 1- 3 


2 3
and
. Prove that an is an integer and bn is an even integer.
A sequence of real numbers a0 , a1 , a2 , … , an , … is defined by a0 = 1 , a1 = 7 and
3.
a n = 3 n +1 - 2
for n = 0 , 1 , 2 , … . Prove , by induction , that
a n +2 - 4a n+1 + 3a n = 0
for all non-negative integers n.
4.
Prove , by induction , that

2n+1 - 
3 +1
2n+1
3 -1
2 n +1 for all positive
is a multiple of
integers n.
5.
A sequence
 a n  is defined by a 2m = - 2m and a 2m-1 = 2m - 1 , where m is a natural number.
Let Sn be the sum of the first n terms of the sequence
6.
Let
u1 = 1 , u2 = 3 and
 a n . Prove that S2m-1 = m
and S2m = - m.
for n  3 .
u n = u n-1 + u n-2
un =  n +  n
Using mathematical induction, or otherwise, prove that
for n  1 ,
where  and  are the roots of x2 - x - 1 = 0 .
7.
Let
 a n  be a sequence of real numbers, where a0 = 1 , a1 = 6 , a2 = 45 and
1
1
a n - a n +1 + a n +2 a
=0
3
27 n +3
(97)
for n = 0 , 1 , 2 , … .


a n = 3 n n 2 + 1 for n = 0 , 1 , 2 , … .
Using mathematical induction, or otherwise, show that
8. Let  and  be the roots of x2 - 14x + 36 = 0. Show that  n +  n is divisible by 2 n for
(98) n = 1 , 2 , 3 , … .
Ex.B
1.
(a)
(79) (b)
Show that if
a  1 and b  1, then
Using mathematical induction, or otherwise, show that for any n positive numbers a1 , a2 , … ,
an , if
a1 a2 … an = 1 , then a1 + a2 + … + an  n , i.e.
a1 a2 … an = 1
(c)

Hence, deduce that if x1 , x2 , … , xn
x1 + x 2 + ... + x n
n
2. (a)
(c)
a1 + a2 + … + an  n .
are n positive numbers, then
(A.M.  G.M.)
 n x1 x 2 ... x n
.
Let a and b be two distinct positive numbers.
Show that for any positive integer n,
(b)
ab + 1  a + b .
Show by induction that,
a n +1 - a n b > a b n - b n +1 .
b n (n + 1) a - nb  < a n +1 .
Using (b) , or otherwise, show that
x

1 + 
n

n
x 

< 1 +

n +1

n +1
for x > 0 .
3.
(96)
4.
(91)