Mathematics - Extended Part (M2)
Past Papers Questions
1. FOUNDATION KNOWLEDGE AREA
1. Mathematical Induction
(1981-HL-GEN MATHS #02) (6 marks)
2. (a) Prove, by mathematical induction, that for any positive integer n ,
1 2
13 + 23 + 33 + . . . + n 3 = n (n + 1)2.
4
(1988-HL-GEN MATHS #07) (8 marks) (Modified)
7. (b) Let An = 12 − 22 + 32 − 42 + . . . + (−1)n−1n 2
n(n + 1)
and Bn = 1 + 2 + 3 + . . . + n =
2
where n is a positive integer.
Show, by mathematical induction, that An = (−1)n−1Bn for all positive integers n .
2m 2m+1
∑ ∑
Hence, or otherwise, find An and An .
n=1 n=1
(1990-HL-GEN MATHS #05) (8 marks)
5. (a) (i) Prove by mathematical induction that for any positive integer n ,
n
1 2
r3 = n (n + 1)2.
∑ 4
r =1
(ii) Find 13 − 23 + 33 − 43 + . . . + (−1)r+1r 3 + . . . − (2n)3 .
(1991-CE-A MATH 2 #07) (8 marks)
7. (a) Prove, by mathematical induction, that
1
12 + 2 2 + . . . + n 2 = n (n + 1)(2n + 1)
6
for all positive integers n .
(b) Using the formula in (a), find the sum of
1 × 2 + 2 × 3 + . . . + n (n + 1).
(1992-CE-A MATH 2 #01) (5 marks)
1. Prove, by mathematical induction, that
1 × 2 + 2 × 5 + 3 × 8 + . . . + n (3n − 1) = n 2(n + 1)
for all positive integers n .
(1993-CE-A MATH 2 #01) (5 marks)
1. Prove that
n (n + 1)(n + 2)(3n + 1)
12 × 2 + 22 × 3 + . . . + n 2(n + 1) =
12
for any positive integer n .
1
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Mathematics - Extended Part (M2)
Past Papers Questions
(1994-CE-A MATH 2 #05) (5 marks)
5. Prove that
1 3 5 2n − 1 2n + 3
+ + + ... + =3−
2 22 23 2n 2n
for any positive integer n .
(1997-CE-A MATH 2 #07) (6 marks)
7. Let Tn = (n 2 + 1)(n!) for any positive integer n . Prove, by mathematical induction, that
T1 + T2 + . . . + Tn = n [(n + 1)!]
for any positive integer n .
(Note: n! = n (n − 1)(n − 2) . . .3 × 2 × 1 )
(1998-CE-A MATH 2 #03) (5 marks)
3. Prove, by mathematical induction, that
1 × 2 + 2 × 3 + 22 × 4 + . . . + 2n−1(n + 1) = 2n (n)
for all positive integers n .
(2000-CE-A MATH 2 #04) (6 marks)
4. Prove, by mathematical induction, that
n (n + 1)
12 − 22 + 32 − 42 + . . . + (−1)n−1n 2 = (−1)n−1
2
for all positive integers n .
(2001-CE-A MATH #12) (8 marks)
12. Prove, by mathematical induction, that
1
1 × 2 + 2 × 3 + 3 × 4 + . . . + n (n + 1) = n (n + 1)(n + 2)
3
for all positive integers n .
Hence evaluate 1 × 3 + 2 × 4 + 3 × 5 + . . . + 50 × 52 .
(2002-CE-A MATH #12) (8 marks)
12. (a) Prove, by mathematical induction, that
2(2) + 3(22 ) + 4(23) + . . . + (n + 1)(2n ) = n (2n+1)
for all positive integers n .
(b) Show that
1(2) + 2(22 ) + 3(23) + . . . + 98(298) = 97(299 ) + 2 .
(2003-CE-A MATH #07) (5 marks)
7. Prove, by mathematical induction, that
1 2 3 n n +2
+ + + ... + n = 2 −
2 2 2 2 3 2 2n
for all positive integers n .
2
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Mathematics - Extended Part (M2)
Past Papers Questions
(2005-CE-A MATH #08) (5 marks)
8. Prove, by mathematical induction, that
1 × 2 2 × 22 3 × 23 n × 2n 2n+1
+ + + = −1
2×3 3×4 4×5 (n + 1)(n + 2) n +2
for all positive integers n .
(2007-CE-A MATH #05) (5 marks)
5. Let a ≠ 0 and a ≠ 1 . Prove by mathematical induction that
1 1 1 1 1
− − − ... − n = n
a −1 a a 2 a a (a − 1)
for all positive integers n .
(2008-CE-A MATH #05) (5 marks)
5. Prove, by mathematical induction, that
1 2
13 + 23 + 33 + . . . + n 3 = n (n + 1)2
4
for all positive integers n .
(2009-CE-A MATH #05) (5 marks)
5. Prove, by mathematical induction, that
1
1 × 4 + 2 × 5 + 3 × 6 + . . . + n (n + 3) = n (n + 1)(n + 5)
3
for all positive integers n .
(2012-DSE-MATH-EP(M2) #03) (5 marks)
3. Prove, by mathematical induction, that for all positive integer n ,
1 × 2 + 2 × 5 + 3 × 8 + . . . + n (3n − 1) = n 2(n + 1).
(2013-DSE-MATH-EP(M2) #03) (5 marks)
3. Prove, by mathematical induction, that for all positive integers n ,
1 1 1 1 4n + 1
1+ + + + ... + = .
1 × 4 4 × 7 7 × 10 (3n − 2) × (3n + 1) 3n + 1
(2016-DSE-MATH-EP(M2) #05) (6 marks)
n
(−1)n n(n + 1)
(−1)k k 2 =
∑
5. (a) Using mathematical induction, prove that for all positive integers n .
k=1
2
333
(−1)k+1k 2 .
∑
(b) Using (a), evaluate
k=3
3
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Mathematics - Extended Part (M2)
Past Papers Questions
(2018-DSE-MATH-EP(M2) #06) (7 marks)
n
n(n + 1)(2n + 13)
∑
6. (a) Using mathematical induction, prove that k (k + 4) = for all positive integers n .
k=1
6
∑ ( 112 ) ( 223 )
555
k k +4
(b) Using (a), evaluate .
k=333
(2019-DSE-MATH-EP(M2) #05) (7 marks)
2n
1 n +1
∑ k (k + 1)
5. (a) Using mathematical induction, prove that = for all positive integers n .
k=n
n (2n + 1)
200
1
∑ k (k + 1)
(b) Using (a), evaluate .
k=50
(2020-DSE-MATH-EP(M2) #05) (7 marks)
n
1 n (n + 3)
∑ k (k + 1)(k + 2)
5. (a) Using mathematical induction, prove that = for all positive
k=1
4(n + 1)(n + 2)
integers n .
123
50
∑ k (k + 1)(k + 2)
(b) Using (a), evaluate .
k=4
(2021-DSE-MATH-EP(M2) #02) (5 marks)
n
n 3(n + 1)3
(3k + k ) =
5 3
∑
2. Using mathematical induction, prove that for all positive integers n .
k=1
2
4
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Mathematics - Extended Part (M2)
Past Papers Questions
ANSWERS
(1991-CE-A MATH 2 #07)
1
7. (b) n (n + 1)(n + 2)
3
(2001-CE-A MATH #12)
12. 45 475
(2016-DSE-MATH-EP(M2) #05)
5. (b) 55 614
(2018-DSE-MATH-EP(M2) #06)
6. (b) 1 813
(2019-DSE-MATH-EP(M2) #05)
151
5. (b)
10050
(2020-DSE-MATH-EP(M2) #05)
387
5. (b)
310
5
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