Download - 11 x1 t14 11 some different types (2012)
![Page 1: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/1.jpg)
nnnn
βαi
tan1tancot1tantan1
that show totantan1
tantantanfact that the Use
2006 Extension 1 HSC Q4d)
Some Different Looking Induction
Problems
![Page 2: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/2.jpg)
nnnn
βαi
tan1tancot1tantan1
that show totantan1
tantantanfact that the Use
nn
nnnn
tan1tan1
tan1tan1tan
2006 Extension 1 HSC Q4d)
Some Different Looking Induction
Problems
![Page 3: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/3.jpg)
nnnn
βαi
tan1tancot1tantan1
that show totantan1
tantantanfact that the Use
nn
nnnn
tan1tan1
tan1tan1tan
2006 Extension 1 HSC Q4d)
nn
nn
tan1tan1
tan1tantan
Some Different Looking Induction
Problems
![Page 4: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/4.jpg)
nnnn
βαi
tan1tancot1tantan1
that show totantan1
tantantanfact that the Use
nn
nnnn
tan1tan1
tan1tan1tan
2006 Extension 1 HSC Q4d)
nn
nn
tan1tan1
tan1tantan
nnnn tan1tantantan1tan1
Some Different Looking Induction
Problems
![Page 5: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/5.jpg)
nnnn
βαi
tan1tancot1tantan1
that show totantan1
tantantanfact that the Use
nn
nnnn
tan1tan1
tan1tan1tan
2006 Extension 1 HSC Q4d)
nn
nn
tan1tan1
tan1tantan
nnnn tan1tantantan1tan1
tan
tan1tantan1tan1
nnnn
Some Different Looking Induction
Problems
![Page 6: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/6.jpg)
nnnn
βαi
tan1tancot1tantan1
that show totantan1
tantantanfact that the Use
nn
nnnn
tan1tan1
tan1tan1tan
2006 Extension 1 HSC Q4d)
nn
nn
tan1tan1
tan1tantan
nnnn tan1tantantan1tan1
tan
tan1tantan1tan1
nnnn
nnnn tan1tancottan1tan1
Some Different Looking Induction
Problems
![Page 7: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/7.jpg)
2006 Extension 1 Examiners Comments;
(d) (i) This part was not done well, with many candidates
presenting circular arguments.
![Page 8: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/8.jpg)
2006 Extension 1 Examiners Comments;
(d) (i) This part was not done well, with many candidates
presenting circular arguments.
However, a number of candidates successfully rearranged
the given fact and easily saw the substitution required.
![Page 9: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/9.jpg)
2006 Extension 1 Examiners Comments;
(d) (i) This part was not done well, with many candidates
presenting circular arguments.
However, a number of candidates successfully rearranged
the given fact and easily saw the substitution required.
Many candidates spent considerable time on this part,
which was worth only one mark.
![Page 10: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/10.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
![Page 11: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/11.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
![Page 12: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/12.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tancot2RHS
![Page 13: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/13.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS 2tancot2RHS
![Page 14: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/14.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS 2tancot2RHS
12tantan1
![Page 15: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/15.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS 2tancot2RHS
12tantan1
1tan2tancot
![Page 16: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/16.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS 2tancot2RHS
12tantan1
1tan2tancot
112tancot
![Page 17: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/17.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS 2tancot2RHS
12tantan1
1tan2tancot
112tancot
22tancot
![Page 18: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/18.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS
Hence the result is true for n = 1
2tancot2RHS
12tantan1
1tan2tancot
112tancot
22tancot
![Page 19: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/19.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
2tancot2RHS
12tantan1
1tan2tancot
112tancot
22tancot
![Page 20: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/20.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
1tancot1
1tantan3tan2tan2tantan ..
kk
kkei
2tancot2RHS
12tantan1
1tan2tancot
112tancot
22tancot
![Page 21: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/21.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
1tancot1
1tantan3tan2tan2tantan ..
kk
kkei
Step 3: Prove the result is true for n = k + 1
2tancot2RHS
12tantan1
1tan2tancot
112tancot
22tancot
![Page 22: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/22.jpg)
1tancot1
1tantan3tan2tan2tantan
1 integers allfor that prove toinduction almathematic Use
nn
nn
nii
Step 1: Prove the result is true for n = 1
2tantanLHS
Hence the result is true for n = 1
Step 2: Assume the result is true for n = k, where k is a positive
integer
1tancot1
1tantan3tan2tan2tantan ..
kk
kkei
Step 3: Prove the result is true for n = k + 1
2tancot2RHS
12tantan1
1tan2tancot
112tancot
22tancot
2tancot2
2tan1tan3tan2tan2tantan ..
kk
kkei
![Page 23: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/23.jpg)
Proof:
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
![Page 24: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/24.jpg)
Proof:
2tan1tan1tancot1 kkkk
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
![Page 25: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/25.jpg)
Proof:
2tan1tan1tancot1 kkkk
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
11tan2tancot1tancot1 kkkk
![Page 26: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/26.jpg)
Proof:
2tan1tan1tancot1 kkkk
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
11tan2tancot1tancot1 kkkk
1tancot2tancot1tancot2 kkkk
![Page 27: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/27.jpg)
Proof:
2tan1tan1tancot1 kkkk
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
11tan2tancot1tancot1 kkkk
1tancot2tancot1tancot2 kkkk
2tancot2 kk
![Page 28: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/28.jpg)
Proof:
2tan1tan1tancot1 kkkk
Hence the result is true for n = k + 1 if it is also true for n = k
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
11tan2tancot1tancot1 kkkk
1tancot2tancot1tancot2 kkkk
2tancot2 kk
![Page 29: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/29.jpg)
Proof:
2tan1tan1tancot1 kkkk
Hence the result is true for n = k + 1 if it is also true for n = k
Step 4: Since the result is true for n= 1, then the result is true for all
positive integral values of n, by induction
2tan1tan1tantan3tan2tan2tantan
2tan1tan3tan2tan2tantan
kkkk
kk
11tan2tancot1tancot1 kkkk
1tancot2tancot1tancot2 kkkk
2tancot2 kk
![Page 30: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/30.jpg)
2006 Extension 1 Examiners Comments;
(ii) Most candidates attempted this part. However, many
only earned the mark for stating the inductive step.
![Page 31: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/31.jpg)
2006 Extension 1 Examiners Comments;
(ii) Most candidates attempted this part. However, many
only earned the mark for stating the inductive step.
They could not prove the expression true for n=1 as they
were not able to use part (i), and consequently not able to
prove the inductive step.
![Page 32: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/32.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
2004 Extension 1 HSC Q4a)
![Page 33: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/33.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
2004 Extension 1 HSC Q4a)
![Page 34: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/34.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
3
1
3
21
LHS
2004 Extension 1 HSC Q4a)
![Page 35: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/35.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
3
1
3
21
LHS
3
1
23
2
RHS
2004 Extension 1 HSC Q4a)
![Page 36: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/36.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
3
1
3
21
LHS
3
1
23
2
RHS
RHSLHS
2004 Extension 1 HSC Q4a)
![Page 37: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/37.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
3
1
3
21
LHS
3
1
23
2
RHS
RHSLHS
Hence the result is true for n = 3
2004 Extension 1 HSC Q4a)
![Page 38: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/38.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
3
1
3
21
LHS
3
1
23
2
RHS
RHSLHS
Hence the result is true for n = 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
2004 Extension 1 HSC Q4a)
![Page 39: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/39.jpg)
1
221
5
21
4
21
3
2-1
;3 integers allfor that prove toinduction almathematic Use
nnn
n
Step 1: Prove the result is true for n = 3
3
1
3
21
LHS
3
1
23
2
RHS
RHSLHS
Hence the result is true for n = 3
Step 2: Assume the result is true for n = k, where k is a positive
integer
2004 Extension 1 HSC Q4a)
122
15
21
4
21
3
2-1 ..
kkkei
![Page 40: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/40.jpg)
Step 3: Prove the result is true for n = k + 1
![Page 41: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/41.jpg)
Step 3: Prove the result is true for n = k + 1
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
![Page 42: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/42.jpg)
Proof:
Step 3: Prove the result is true for n = k + 1
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
![Page 43: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/43.jpg)
Proof:
Step 3: Prove the result is true for n = k + 1
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
1
21
1
2
kkk
![Page 44: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/44.jpg)
Proof:
Step 3: Prove the result is true for n = k + 1
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
1
21
1
2
kkk
1
21
1
2
k
k
kk
![Page 45: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/45.jpg)
Proof:
Step 3: Prove the result is true for n = k + 1
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
1
21
1
2
kkk
1
21
1
2
k
k
kk
1
1
1
2
k
k
kk
![Page 46: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/46.jpg)
Proof:
Step 3: Prove the result is true for n = k + 1
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
1
21
1
2
kkk
1
21
1
2
k
k
kk
1
1
1
2
k
k
kk
1
2
kk
![Page 47: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/47.jpg)
Proof:
Step 3: Prove the result is true for n = k + 1
Step 4: Since the result is true for n = 3, then the result is true for
all positive integral values of n > 2 by induction.
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
1
21
1
2
kkk
1
21
1
2
k
k
kk
1
1
1
2
k
k
kk
1
2
kk
![Page 48: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/48.jpg)
Proof:
Hence the result is true for
n = k + 1 if it is also true
for n = k
Step 3: Prove the result is true for n = k + 1
Step 4: Since the result is true for n = 3, then the result is true for
all positive integral values of n > 2 by induction.
kkkei
1
2
1
21
5
21
4
21
3
2-1 Prove ..
1
21
21
5
21
4
21
3
2-1
1
21
5
21
4
21
3
2-1
kk
k
1
21
1
2
kkk
1
21
1
2
k
k
kk
1
1
1
2
k
k
kk
1
2
kk
![Page 49: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/49.jpg)
2004 Extension 1 Examiners Comments;
(a) Many candidates submitted excellent responses
including all the necessary components of an
induction proof.
![Page 50: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/50.jpg)
2004 Extension 1 Examiners Comments;
(a) Many candidates submitted excellent responses
including all the necessary components of an
induction proof.
Weaker responses were those that did not verify for n = 3,
had an incorrect statement of the assumption for n = k or
had an incorrect use of the assumption in the attempt to
prove the statement.
![Page 51: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/51.jpg)
2004 Extension 1 Examiners Comments;
(a) Many candidates submitted excellent responses
including all the necessary components of an
induction proof.
Weaker responses were those that did not verify for n = 3,
had an incorrect statement of the assumption for n = k or
had an incorrect use of the assumption in the attempt to
prove the statement.
Poor algebraic skills made it difficult or even impossible for
some candidates to complete the proof.
![Page 52: 11 x1 t14 11 some different types (2012)](https://reader033.vdocuments.net/reader033/viewer/2022060116/558417f2d8b42a34708b4e42/html5/thumbnails/52.jpg)
2004 Extension 1 Examiners Comments;
(a) Many candidates submitted excellent responses
including all the necessary components of an
induction proof.
Weaker responses were those that did not verify for n = 3,
had an incorrect statement of the assumption for n = k or
had an incorrect use of the assumption in the attempt to
prove the statement.
Poor algebraic skills made it difficult or even impossible for
some candidates to complete the proof.
Rote learning of the formula S(k) +T(k+1) = S(k+1) led
many candidates to treat the expression as a sum rather
than a product, thus making completion of the proof
impossible.