F.
4 Mathematics M2 – Mathematical Induction / Binomial Theorem (Public Exam Questions)
1 𝑛(𝑛+3)
1. (a) Using mathematical induction, prove that ∑𝑛𝑘=1 𝑘(𝑘+1)(𝑘+2) = 4(𝑛+1)(𝑛+2) for all
positive integers 𝑛.
1
(b) Using (a), evaluate ∑123
𝑘=4 . 2020 #5 [7]
𝑘(𝑘+1)(𝑘+2)
1 𝑛+1
2. (a) Using mathematical induction, prove that ∑2𝑛
𝑘=𝑛 𝑘(𝑘+1) = 𝑛(2𝑛+1) for all positive
integers 𝑛.
1
(b) Using (a), evaluate ∑200
𝑘=50 𝑘(𝑘+1). 2019 #5 [7]
𝑛(𝑛+1)(2𝑛+13)
3. (a) Using mathematical induction, prove that ∑𝑛𝑘=1 𝑘(𝑘 + 4) = for all
6
positive integers 𝑛.
𝑘 𝑘+4
(b) Using (a), evaluate ∑555
𝑘=333 (112) ( 223 ). 2018 #6 [7]
(−1)𝑛𝑛(𝑛+1)
4. (a) Using mathematical induction, prove that ∑𝑛𝑘=1(−1)𝑘 𝑘 2 = for all positive
2
integers 𝑛.
(b) Using (a), evaluate ∑333
𝑘=3(−1) 𝑘 .
𝑘+1 2
2016 #5 [6]
𝑥 𝑛𝑥 (𝑛+1)𝑥
5. (a) Using mathematical induction, prove that sin 2 ∑𝑛𝑘=1 cos 𝑘𝑥 = sin 2 cos 2 for all
positive integers 𝑛.
𝑘𝜋
(b) Using (a), evaluate ∑567
𝑘=1 cos . 2015 #8 [8]
7
6. Prove, by mathematical induction, that for all positive integers 𝑛, 2013 #3 [5]
1 1 1 1 4𝑛 + 1
1+ + + + ⋯+ = .
1 × 4 4 × 7 7 × 10 (3𝑛 − 2)(3𝑛 + 1) 3𝑛 + 1
7. Prove, by mathematical induction, that for all positive integers 𝑛, 2012 #3 [5]
1 × 2 + 3 × 5 + 3 × 8 + ⋯ + 𝑛(3𝑛 − 1) = 𝑛2 (𝑛 + 1).
8. Prove by mathematical induction, that 4𝑛 + 15𝑛 − 1 is divisible by 9 for all positive
integers 𝑛. DSE Practice #3 [5]
9. (a) Expand (1 − 𝑥)4 .
(b) Find the constant 𝑘 such that the coefficient of 𝑥 2 in the expansion of
(1 + 𝑘𝑥)9 (1 − 𝑥)4 is −3. 2020 #1 [4]
10. Expand (𝑥 + 3)5 . Hence find the coefficient of 𝑥 3 in the expansion of
4 2
(𝑥 + 3)5 (𝑥 − ) . 2018 #2 [5]
𝑥
11. Let (1 + 𝑎𝑥)8 = ∑8𝑘=0 𝜆𝑘 𝑥 𝑘
and (𝑏 + = 𝑥)9 ∑9𝑘=0 𝜇𝑘 𝑥 𝑘 ,
where 𝑎 and 𝑏 are constants. It is
given that 𝜆2 : 𝜇7 = 7: 4 and 𝜆1 + 𝜇8 + 6 = 0. Find 𝑎. 2017 #2 [5]
12. Expand (5 + 𝑥) . Hence find the constant term in the expansion of
4
2 3
(5 + 𝑥)4 (1 − ) . 2016 #1 [5]
𝑥
13. In the expansion of (1 − 4𝑥) 2 (1 + 𝑥)𝑛 , the coefficient of 𝑥 is 1. 2014 #1 [4]
(a) Find the value of 𝑛.
(b) Find the coefficient of 𝑥 2 .
14. Suppose the coefficients of 𝑥 and 𝑥 2 in the expansion of (1 + 𝑎𝑥)𝑛 are −20 and 180
respectively. Find the values of 𝑎 and 𝑛. 2013 #2 [4]
15. It is given that
(1 + 𝑎𝑥)𝑛 = 1 + 6𝑥 + 16𝑥 2 + terms involving higher powers of 𝑥,
where 𝑛 is a positive integer and 𝑎 is a constant. Find the values of 𝑎 and 𝑛. 2012 #2 [5]
16. Find the coefficient of 𝑥 in the expression of (2 − 𝑥) .
5 9 DSE Practice #1 [4]
