Date | May 2018 | Marks available | 7 | Reference code | 18M.2.hl.TZ2.6 |
Level | HL only | Paper | 2 | Time zone | TZ2 |
Command term | Use and Prove that | Question number | 6 | Adapted from | N/A |
Question
Use mathematical induction to prove that (1−a)n>1−na for {n:n∈Z+,n⩾2} where 0<a<1.
Markscheme
Let Pn be the statement: (1−a)n>1−na for some n∈Z+,n⩾2 where 0<a<1 consider the case n=2:(1−a)2=1−2a+a2 M1
>1−2a because a2<0. Therefore P2 is true R1
assume Pn is true for some n=k
(1−a)k>1−ka M1
Note: Assumption of truth must be present. Following marks are not dependent on this M1.
EITHER
consider (1−a)k+1=(1−a)(1−a)k M1
>1−(k+1)a+ka2 A1
>1−(k+1)a⇒Pk+1 is true (as ka2>0) R1
OR
multiply both sides by (1−a) (which is positive) M1
(1−a)k+1>(1−ka)(1−a)
(1−a)k+1>1−(k+1)a+ka2 A1
(1−a)k+1>1−(k+1)a⇒Pk+1 is true (as ka2>0) R1
THEN
P2 is true Pk is true ⇒Pk+1 is true so Pn true for all n>2 (or equivalent) R1
Note: Only award the last R1 if at least four of the previous marks are gained including the A1.
[7 marks]