F
F.6 Pure Mathematics Quiz 02
Total score: 34 Time allowed: 60 minutes
1. Evaluate
(a) [pic].
(b) [pic]. [Hint: k2 ( 1 ( (k ( 1)(k + 1)]
(6 marks)
2. (a) Show that [pic] for any natural number k > 1.
(b) Hence or otherwise, show that [pic] < 2 for any natural number n.
(7 marks)
3. Let {an} be a sequence of numbers such that a0 = 0, a1 = 1, 6an = 5an ( 1 ( an ( 2 for n ( 2.
Show that an = 6[pic] for any natural number n.
(8 marks)
4. By using mathematical induction or otherwise, prove that 1 +[pic] + [pic] + ... +[pic] > [pic]
for any natural number n.
(6 marks)
5. By using mathematical induction, show that [pic]
for any natural number n.
[Hint: you may use the fact that 12 + 22 + … + n2 = [pic]]
(7 marks)
END OF PAPER
Marking Scheme of F.6 Pure Mathematics Quiz 02
1.
(a) [pic] = [pic] 1M
= [pic] + [pic] = [pic] = [pic] 1M+1A
(b) [pic] = [pic] = [pic] 1M
= [pic] = [pic] = [pic] 1M+1A
2.
(a) ∵ [pic] = [pic] 1M+1A
( [pic] 1
(b) [pic] < [pic] 1M+1A
( [pic] < [pic] = 1 ( [pic] 1M
( [pic] < 2 ( [pic] < 2 1M
3. When n = 0, 6[pic] = 6(1 ( 1) = 0 ( proposition is true for n = 0.
When n = 1, 6[pic] = 6[pic] = 1 ( proposition is true for n = 1. 2A
Suppose ak = 6(( k ( ( k) and ak + 1 = 6(( k + 1 ( ( k + 1) ; where ( = [pic], ( = [pic] 1
Consider ak + 2 = [pic]= [pic] 1
= (k(5( ( 1) ( (k(5( ( 1) 1M
= (k([pic]) ( (k([pic]) [∵ ( = [pic], ( = [pic]] 1M
= 6[(k([pic])2 ( (k([pic])2] [∵ ( = [pic], ( = [pic]]
= 6(( k + 2 ( ( k + 2) [∵ ( = [pic], ( = [pic]] 1
By the principle of mathematical induction, an = 6[pic] is true for any non-negative
integer n. 1
4. When n = 1, L.H.S. = 1; R.H.S. = [pic], thus, the inequality is true for n = 1 1A
Suppose 1 +[pic] + [pic] + ... +[pic] > [pic] 1M
Consider 1 +[pic] + [pic] + ... +[pic] + ([pic] + [pic] + … + [pic]) 1A
> [pic] + ([pic] + [pic] + … +[pic]) [M.I. assumption] 1M
> [pic] + ([pic]+[pic]+ … +[pic]) 1M
= [pic] + [pic] = [pic]
Hence, the inequality is true for n = k + 1, by the principle of mathematical induction, the
inequality is true for any natural number n. 1
5. When n = 1, L.H.S. = [pic] = 1; R.H.S. = [pic] = 1 1A
Hence the equation holds for n = 1.
Suppose [pic] 1M
Consider
[pic]
= [pic] 1M
= [pic] = [pic]
= [pic] 1M
= [pic] + [pic]
= [pic] + [pic] 1M
= [pic]
= [pic] = [pic] 1M
= [pic] = [pic] 1
By the principle of mathematical induction, the equation is valid for any natural number n.
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