0% found this document useful (0 votes)

36 views56 pages

Series and Proofs Questions 0525

The document contains a series of mathematical problems related to sequences, series, and mathematical induction, with varying maximum marks assigned to each problem. Topics include arithmetic and geometric sequences, sums of series, and proofs using induction. Each section outlines specific tasks such as finding sums, determining terms, and proving properties of sequences.

Uploaded by

Venkatesh Sundararajan

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

0% found this document useful (0 votes)

36 views56 pages

Series and Proofs Questions 0525

Uploaded by

Venkatesh Sundararajan

We take content rights seriously. If you suspect this is your content, claim it here.

Available Formats

Download as PDF, TXT or read online on Scribd

You are on page 1/ 56

D

6)
=

9
10)

)
12)

14)

15)

16)
18)

(a)

2)
2. [Maximum mark: 6]
(a) Find the value of sum
39 + 48 + 57 + 66 + L + 246. [3 marks]
n
(b) This sum can be expressed in the form ∑ (ak + b) .
k =1

Find the values of a, b and n. [3 marks]

.....................................................................

2
3. [Maximum mark: 6]
Consider the infinite geometric series
1 1 3
− 2
+ +L
3k 3 6k 12k 3
(a) Find the possible values of k given that the series converges. [4 marks]
(b) Find the exact value of the infinite sum for k = 1. [2 marks]

.....................................................................

3
4. [Maximum mark: 7]
In an arithmetic sequence the sum of the first six terms is 75 and the sum of
the first ten terms is 185.
(a) Find the sum of the first eight terms. [5 marks]
(b) Find the number of terms which are less than 1000. [2 marks]

.....................................................................

4
5. [Maximum mark: 5]
The third term of an infinite geometric sequence is –108 and the fifth term is –72.
Find the possible values for the sum to infinity of the sequence.

.....................................................................

5
6. [Maximum mark: 7]
The value of a share decreases by 2% each day. Its value today is 680€.
Find
(a) the value of the share after 9 days. [2 marks]
(b) the value of the share 10 days ago. [2 marks]
(c) after how many days the value of the share will fall below 400€ [3 marks]

.....................................................................

6
7. [Maximum mark: 8]
(a) In an arithmetic sequence, the sum of the first ten terms is equal to the tenth
term. Find S 9 and u 5 . [5 marks]

(b) The third term of an arithmetic sequence is 7. Find S 5 [3 marks]

.....................................................................

7
8. [Maximum mark: 6]
The sum of the first n terms of a series is given by
S n = n 3 + n where n ∈ +.
(a) Find the first three terms of the sequence and show that it is
neither arithmetic nor geometric. [4 marks]

(b) Find the 10th term of the sequence. [2 marks]

.....................................................................

8
9. [Maximum mark: 8]
(a) Consider the expansion of
8
 3 3
x − 
 x
Find
(i) Find the constant term.
(ii) Find the term in x 4 . [5 marks]

(b) Consider now

8
 3 3
 x −  3x − 2
2
( )
2

 x
Find the term in x 4 . [3 marks]

.....................................................................

.....................................................................
9
10. [Maximum mark: 6]
In the expansion of
n
 3
x 3 1 − 
 x
the constant term is – 3240. Find
(a) the value of n. [4 marks]
(b) the coefficient of x. [2 marks]

.....................................................................

.....................................................................
10
11. [Maximum mark: 8]
The distinct numbers
x + 9y , x + 3y , x
are the first three terms of a geometric sequence.
(a) Find the common ratio. [5 marks]
(b) Given that the sum to infinity is 48, find the values of x and y. [3 marks]

.....................................................................

11
12. [Maximum mark: 12]
1 2 3 n
Let Sn = + + + L +
2! 3! 4! (n + 1)!
(a) Calculate S1 , S 2 , S 3 , S 4 [2 marks]
(b) Write down the values of 1!, 2!, 3!, 4!, 5! [2 marks]
(c) Hence, guess a formula for S n , for any n ∈ Z + [1 mark]
(d) Show by using mathematical induction that your guess is true. [7 marks]

.....................................................................

12
13. [Maximum mark: 15]
(a) Express ( 3+ 2 ) 3
in the form a 2 + b 3 where a, b ∈ Z + [2 marks]
(b) Express ( 2) ( )
2 4
3+ and 3+ 2 in the form a + b 6 ,
+
where a, b ∈ Z [3 marks]
+
(c) By using mathematical induction, show that, for any n ∈ Z
( 3+ 2 )
2n
has the form a + b 6 , where a, b ∈ Z + [7 marks]
(d) Hence show that, for any n ∈ Z +
( 3+ 2 )
2 n +1
has the form a 2 + b 3 , where a, b ∈ Z + [3 marks]

.....................................................................

14
2. [Maximum mark: 5]
Consider the infinite geometric series
r
∞
9  3k + 1 
∑ 
r =1 2  3 


(a) Find the possible values of k given that the series converges. [3 marks]
(b) Find the exact value of the infinite sum for k = – 1. [2 marks]

.....................................................................

2
3. [Maximum mark: 6]
In an arithmetic sequence the sixth term is 95 while the sum of the first 21 terms is
3360. Find
(a) the first term which exceeds 1000. [4 marks]
(b) the sum of the terms which are less than 1000. [2 marks]

.....................................................................

3
4. [Maximum mark: 6]
The population of ants in a field today, 10 October 2017, is 120,000 and increases
by 9% each day.
(a) Find the population of ants on 31 October 2017. [3 marks]
This population at some time exceeds 1,000,000.
(b) Write down an inequality to represent this information for the number
of days needed and hence find the date when it will happen. [3 marks]

.....................................................................

4
5. [Maximum mark: 6]
The integers a, b and b+2a are consecutive terms in an arithmetic sequence.
The integers a, 20 and a+5b are consecutive terms in a geometric sequence.
Find the values of a and b.

.....................................................................

5
6. [Maximum mark: 7]
(a) In an arithmetic sequence, the sum of the first ten terms is equal to the sum of
the first 9 terms. Show that
u16 = −u4 [3 marks]
(b) Find the sum of the numbers which are multiples of 8 but not multiples of 6,
between 100 and 3100. [4 marks]

.....................................................................

6
7. [Maximum mark: 6]
The sum of the first n terms of a series is given by
S n = 5n +1 − 5 where n ∈ +.
(a) Find the fifth term of the sequence. [2 marks]
(b) Find the n-th term of the sequence in terms of n. [2 marks]
(c) Show that the sequence is geometric. [2 marks]

.....................................................................

7
8. [Maximum mark: 7]
(a) Find the constant term in the expansion of
10
 3 3
x − 2  [3 marks]
 x 
(b) Find the constant term in the expansion of
10
 3 3 1 
 x − 2   5 + x + 2
3
[4 marks]
 x  x 

.....................................................................

8
9. [Maximum mark: 6]
In the expansion of
(2a + ax ) n
the coefficient of x 4 is five times the coefficient of x 2 . Find the value of n.

.....................................................................

9
10. [Maximum mark: 8]
The numbers a, b and c are the second, the fifth and the eleventh term of an
arithmetic sequence. They are also consecutive terms in an increasing geometric
sequence.
(a) Find the common ratio of the geometric sequence. [5 marks]
(b) Given that the third term of the arithmetic sequence is 20, find the values of
a, b and c . [3 marks]

.....................................................................

10
11. [Maximum mark: 7]
Using mathematical induction, prove that
the number 2 2 n +1 − 6n + 16 is divisible by 18, for any n ∈ Z +

.....................................................................

11
12. [Maximum mark: 10]
(
(a) Express 1+ 2 3 ) 3
in the form a + b 3 where a, b ∈ Z + [2 marks]
(b) By using mathematical induction, show that for any n ∈ Z + ,
(1 + 2 3 ) n
has the form a + b 3
where a is an odd integer and b is an even integer. [8 marks]

.................................. ...................................

.....................................................................

.....................................................................
12
2. [Maximum mark: 9]
(a) Prove by mathematical induction that [7]
n

∑r ⋅2
r =2
r
= (n − 1)2 n +1 for any integer n ≥ 2 .

(b) Confirm that the statement is true for n = 3 . [2]

..................................................................................................................................

2
3. [Maximum mark: 7]
Prove by mathematical induction that
(n + 1)!≥ 2 n n for any integer n ≥ 3 .

..................................................................................................................................

..................................................................................................................................
Turn over

3
4. [Maximum mark: 8]
A sequence is defined recursively as follows
u1 = 1
u n +1 = 3u n +1
(a) Write down the first four terms of the sequence. [2]
(b) Prove by mathematical induction that
3n − 1
un = for any n = 1,2,3,... . [6]
2

..................................................................................................................................

4
5. [Maximum mark: 10]

(a) Expand ( 2 + 5 ) 2 . [1]

(b) Prove by mathematical induction that ( 2 + 5 ) 2 n has the form

a + b 10 , where a is an odd integer and b is an even integer.

for any n ∈ Z + . [7]

(c) Hence, show that ( 2 + 5 ) 2 n +1 has the form

c 2 + d 5 , where c, d ∈ Z for any n ∈ Z +

without using mathematical induction. [2]

..................................................................................................................................

..................................................................................................................................
Turn over

5
2. [Maximum mark: 9]
(a) Prove by mathematical induction that [7]
1 ⋅ 21 + 2 ⋅ 2 2 + 3 ⋅ 2 3 + L + n ⋅ 2 n = (n − 1)2 n +1 + 2 for any n ∈ Z + .

(b) Confirm that the statement is true for n = 3 . [2]

..................................................................................................................................

2
3. [Maximum mark: 7]
Prove by mathematical induction that
3 n > n 2 + 2n for any integer n ≥ 2 .

..................................................................................................................................

..................................................................................................................................
Turn over

3
4. [Maximum mark: 8]
A sequence is defined recursively as follows
u1 = 5
u n +1 = 2u n +5
(a) Write down the first four terms of the sequence. [2]
(b) Prove by mathematical induction that
un = 5 ⋅ 2n − 5 for any n = 1,2,3,... . [6]

..................................................................................................................................

4
5. [Maximum mark: 10]
(a) Prove by mathematical induction that
n!×[2 × 6 × 10 × 14 × L × (4n − 2)] = (2n)! for any n ∈ Z + . [8]

a!
(b) Express (2 × 6 × 10 × 14 × L × 78) in the form . [2]
b!

..................................................................................................................................

..................................................................................................................................
5
2. [Maximum mark: 9]
It is claimed that
1  21  2  2 2  3  2 3  ⋯  n  2 n  ( n  1) 2 n 1  2 for any n  Z  .

(a) Show that the claim is true for n  4 [2]

(b) Prove the claim by using mathematical induction. [7]

..................................................................................................................................

2
3. [Maximum mark: 6]
Prove by mathematical induction that
(2n)!  2n n for any integer n  2 .

..................................................................................................................................

..................................................................................................................................
Turn over

3
4. [Maximum mark: 9]
(a) Prove, by using contradiction, that for a, b  ℝ  {2}

2a  3 2b  3
if a  b then  [5]
a2 b2
(b) State if the following statement is also true:
2a 2  3 2b 2  3
if a  b then  2
a2  2 b 2
Justify your answer, either by giving a proof, or by using a counterexample. [2]
(c) Based on the results in (a) and (b), write down a conclusion for the functions

2x  3 2 x2  3
f ( x)  , x  2 and g ( x )  2 . [2]
x2 x 2

..................................................................................................................................

4
2. [Maximum mark: 6]
Given that
(a − 3 ) 4
= 97 − b 3

where a and b are positive integers, find the value of a and of b. [6 marks]

.....................................................................

Page 2
3. [Maximum mark: 7]
Given that
(2 + x) 5 (1 + 2 x) n = 32 + ax + 27600 x 2 + L + 2 n x n + 5
find the values of the integers a and n. [7 marks]

.....................................................................

Page 3
4. [Maximum mark: 11]
Solve analytically the equations

(a) log 3 ( x − 3) = 1 + log 9 ( x + 1) (b) log x 2 − 3 log 2 x = 2 [6+5 marks]

.....................................................................

Page 4
5. [Maximum mark: 11]
Solve the exponential equations
3 ⋅ (5 x ) ln a
(a) 2 x +5 = x − 2 . Give your answer in the form x = , where a, b ∈ Q . [5 marks]
2 ln b
3
(b) 2 x +5 = 8 + . Give your answer in the form x = a + log 2 b , where a, b ∈ Z . [6 marks]
2 x −2

.....................................................................

Page 5
6. [Maximum mark: 9]
Solve the equations
x ln x x [4+5 marks]
(a) x log x = 10000 (b) 2
=
x e

.....................................................................

Page 6
7. [Maximum mark: 6]
x
It is given that log a ( x 2 y ) = 14 and log a ( )=2
y2
(a) Find log a x and log a y
[5 marks]
(b) Hence find log y x [1 mark]

.....................................................................

Page 7
8. [Maximum mark: 6]

Solve the simultaneous equations:

log 2 x − log 4 y = 4
log 2 ( x − 2 y ) = 7 log 7 5

.....................................................................

Page 8
9. [Maximum mark: 11]
Solve the equations
(a) ln x 2 + ln x = 2 [3 marks]
(b) (ln x) 2 + ln x = 2 [4 marks]
(c) (ln x)(ln x)(ln x) = ln x 9 [4 marks]

.....................................................................

Page 9
10. [Maximum mark: 9]

(a) Prove by mathematical induction that 2 3n − 3 n is divisible by 5, for any n ≥ 1 . [7 marks]

(b) Hence find the last digit of the number 2( 2 6045 − 3 2015 ) [2 marks]

.....................................................................

Page 10
11. [Maximum mark: 12]
(a) Prove by using mathematical induction that
0 1 2 n − 1 n!−1
+ + +L+ = , for any n ≥ 1 . [7 marks]
1! 2! 3! n! n!
6
r −1 1
(b) Given that ∑
r =1 r!
= 1 − find the value of a.
a [2 marks]

n
r −1
(c) Find the smallest value of n so that ∑
r =1 r!
> 0.99999 [2 marks]
n
r −1
(d) Find the largest value of n so that ∑
r =1 r!
< 0.99999 [1 mark]

.....................................................................

.....................................................................
Page 11
12. [Maximum mark: 8]
(a) Given the property
ln x + ln y = ln xy
prove by using mathematical induction that
n ln x = ln x n , for any positive integer n. [7 marks]
(b) Is the property true for n = 0 ? [1 mark]

.....................................................................

Page 13

IB Math HL: Sequences & Series
No ratings yet
IB Math HL: Sequences & Series
15 pages
TEST 5. Trigonometry (P1-P2) (2016)
No ratings yet
TEST 5. Trigonometry (P1-P2) (2016)
12 pages
TEST 2. Trigonometry (P1-P2) (2013)
No ratings yet
TEST 2. Trigonometry (P1-P2) (2013)
8 pages
REVISION 1 (Logarithms - Binomial - Induction)
No ratings yet
REVISION 1 (Logarithms - Binomial - Induction)
13 pages
TEST 3. Trigonometry (2014)
No ratings yet
TEST 3. Trigonometry (2014)
10 pages
Trigonometry HW 3
No ratings yet
Trigonometry HW 3
7 pages
TEST 5. Trigonometry (P1-P2) (2016)
No ratings yet
TEST 5. Trigonometry (P1-P2) (2016)
16 pages
P1 Mock Exam Math Analysis HL
No ratings yet
P1 Mock Exam Math Analysis HL
12 pages
Math HL Polynomials Test
No ratings yet
Math HL Polynomials Test
13 pages
Integration 1
No ratings yet
Integration 1
12 pages
AA Mock Paper 2
No ratings yet
AA Mock Paper 2
6 pages
Math SL Exponents & Logs Test
No ratings yet
Math SL Exponents & Logs Test
8 pages
Functions IB Test
No ratings yet
Functions IB Test
87 pages
TEST 6. Trigonometry 1 (P1-P2) (2018)
No ratings yet
TEST 6. Trigonometry 1 (P1-P2) (2018)
21 pages
P2 Mock Exam Math Analysis HL 2023-2024
No ratings yet
P2 Mock Exam Math Analysis HL 2023-2024
12 pages
TEST 4. Triangles - Arcs and Sectors (2016)
No ratings yet
TEST 4. Triangles - Arcs and Sectors (2016)
8 pages
Revision 2 P1 (2016)
No ratings yet
Revision 2 P1 (2016)
10 pages
TEST 7. Trigonometry 2 (2018)
No ratings yet
TEST 7. Trigonometry 2 (2018)
13 pages
OPERATION OF POLYNOMIALS Copy
No ratings yet
OPERATION OF POLYNOMIALS Copy
12 pages
TEST 3. Derivatives (2014)
No ratings yet
TEST 3. Derivatives (2014)
12 pages
Mathematics Tut # 6A: Hexagon
No ratings yet
Mathematics Tut # 6A: Hexagon
8 pages
(2024) AAHL Set B Paper 1 v3
No ratings yet
(2024) AAHL Set B Paper 1 v3
14 pages
Exam 250609 231036
No ratings yet
Exam 250609 231036
3 pages
Mathematics Tut # 5A: Hexagon
No ratings yet
Mathematics Tut # 5A: Hexagon
6 pages
Math 307 Quiz: Matrix Inverses
No ratings yet
Math 307 Quiz: Matrix Inverses
2 pages
Institute of Mathematics College of Science University of The Philippines Diliman, Quezon City
No ratings yet
Institute of Mathematics College of Science University of The Philippines Diliman, Quezon City
12 pages
Integrals (2013)
No ratings yet
Integrals (2013)
12 pages
Revision 2 p2 (2016)
No ratings yet
Revision 2 p2 (2016)
8 pages
Paper 1 Practice Set B
No ratings yet
Paper 1 Practice Set B
11 pages
IB Math SL Trigonometry Exam
No ratings yet
IB Math SL Trigonometry Exam
6 pages
Test 2. Functions - Equations
No ratings yet
Test 2. Functions - Equations
11 pages
A2 2022
No ratings yet
A2 2022
22 pages
Linear Algebra Lec 1 HND
No ratings yet
Linear Algebra Lec 1 HND
21 pages
Math Jan Mock Paper 1
No ratings yet
Math Jan Mock Paper 1
21 pages
Class Activity 2 - LCM MCD Order or Operation Demonstration-1
No ratings yet
Class Activity 2 - LCM MCD Order or Operation Demonstration-1
8 pages
P2 Mock Exam Math Analysis HL
No ratings yet
P2 Mock Exam Math Analysis HL
14 pages
WJEC GCSE Higher P1 S2008
No ratings yet
WJEC GCSE Higher P1 S2008
25 pages
Aa HL Paper 1 May 2022 Tz1
No ratings yet
Aa HL Paper 1 May 2022 Tz1
12 pages
Final 2019
No ratings yet
Final 2019
7 pages
PM2002 Quiz 2
No ratings yet
PM2002 Quiz 2
2 pages
Ilovepdf - Merged (60) - Removed
No ratings yet
Ilovepdf - Merged (60) - Removed
9 pages
(2024) AAHL Set B Paper 2 v3
No ratings yet
(2024) AAHL Set B Paper 2 v3
18 pages
AA SL Nov 2022 P1
No ratings yet
AA SL Nov 2022 P1
3 pages
Practice Questions
No ratings yet
Practice Questions
9 pages
Writing Task 2: Write About The Following Topic
No ratings yet
Writing Task 2: Write About The Following Topic
5 pages
Pcbnew
No ratings yet
Pcbnew
129 pages
Pcbnew
No ratings yet
Pcbnew
162 pages
Practice Paper 1A (60 Marks 60 Mins)
No ratings yet
Practice Paper 1A (60 Marks 60 Mins)
6 pages
Writing Task 2: Name: Date
No ratings yet
Writing Task 2: Name: Date
5 pages
Getting Started in Kicad
No ratings yet
Getting Started in Kicad
53 pages
Getting Started in KiCad
No ratings yet
Getting Started in KiCad
54 pages
KiCad Beginner's Guide
No ratings yet
KiCad Beginner's Guide
52 pages
KiCad 60 Getting Started
No ratings yet
KiCad 60 Getting Started
54 pages
Kit Arch Studyguide 221124
No ratings yet
Kit Arch Studyguide 221124
140 pages
Lesson 17
No ratings yet
Lesson 17
4 pages
2024 2 TZ2 HL
No ratings yet
2024 2 TZ2 HL
14 pages
Eeschema
No ratings yet
Eeschema
235 pages
Distribution Worksheet
No ratings yet
Distribution Worksheet
8 pages
Kinetic Theory of Gasses-Formulas
No ratings yet
Kinetic Theory of Gasses-Formulas
3 pages
Motion in 1D and Projectiles
No ratings yet
Motion in 1D and Projectiles
10 pages
IB-HL Paper 3 Questions
No ratings yet
IB-HL Paper 3 Questions
45 pages
Kinematics Qns 050625
No ratings yet
Kinematics Qns 050625
8 pages
Vectors Additional Questions
No ratings yet
Vectors Additional Questions
23 pages
Binomial, Induction and Sequences Practice
No ratings yet
Binomial, Induction and Sequences Practice
68 pages
Final 2023
No ratings yet
Final 2023
2 pages
Physics Practice Test 160424 Answers Marked
No ratings yet
Physics Practice Test 160424 Answers Marked
55 pages
Bitsat Integration Solutions
No ratings yet
Bitsat Integration Solutions
8 pages
Bitsat Indef Integrals Qns
No ratings yet
Bitsat Indef Integrals Qns
4 pages
Find A Formula or Conjecture For Each of The Patterns Shown Below, and Give The Next Three Terms of Each Pattern
No ratings yet
Find A Formula or Conjecture For Each of The Patterns Shown Below, and Give The Next Three Terms of Each Pattern
48 pages
R S Aggarwal Solutions Class 11 Maths Chapter 15 Trigonometric or Circular Functions
100% (1)
R S Aggarwal Solutions Class 11 Maths Chapter 15 Trigonometric or Circular Functions
131 pages
Algebra Concepts for Students
100% (1)
Algebra Concepts for Students
46 pages
RISE QM MCQs Ch#03
No ratings yet
RISE QM MCQs Ch#03
12 pages
Department of Education Region XII Division of South Cotabato Polomolok Creek Integrated School Brgy. Magsasay, Polomolok, South Cotabato
No ratings yet
Department of Education Region XII Division of South Cotabato Polomolok Creek Integrated School Brgy. Magsasay, Polomolok, South Cotabato
1 page
Review Sheet - GR10
No ratings yet
Review Sheet - GR10
2 pages
Global Tour
No ratings yet
Global Tour
9 pages
Unit 7 Sequences and Series Unit Plan
No ratings yet
Unit 7 Sequences and Series Unit Plan
6 pages
Population & Water Demand Study
67% (3)
Population & Water Demand Study
32 pages
Notes
No ratings yet
Notes
22 pages
Grade XI Math Homework Tasks
No ratings yet
Grade XI Math Homework Tasks
16 pages
Geometric Sequences Explained
No ratings yet
Geometric Sequences Explained
6 pages
4027 Pure Mathematics CALA B
No ratings yet
4027 Pure Mathematics CALA B
3 pages
Mathematics ÔÇó Final Step-A ÔÇó VMC
No ratings yet
Mathematics ÔÇó Final Step-A ÔÇó VMC
57 pages
9709 w09 QP 11
No ratings yet
9709 w09 QP 11
4 pages
Mathematics10 Quarter1 Module4 Week4
No ratings yet
Mathematics10 Quarter1 Module4 Week4
6 pages
First Quarter Summative Assessment
No ratings yet
First Quarter Summative Assessment
4 pages
AP & GP Assignment, IB 1 Assignment, 13th May, 2022, PDF
No ratings yet
AP & GP Assignment, IB 1 Assignment, 13th May, 2022, PDF
21 pages
Sequence Finite Sequence Infinite Sequence Series: For Example
No ratings yet
Sequence Finite Sequence Infinite Sequence Series: For Example
2 pages
HSSC-1 Mathematics Topics
No ratings yet
HSSC-1 Mathematics Topics
17 pages
Activity 2 Geometric Sequence
No ratings yet
Activity 2 Geometric Sequence
2 pages
Grade 10 Math Test
100% (1)
Grade 10 Math Test
2 pages
Grade 10 Math Exam: Sequences
No ratings yet
Grade 10 Math Exam: Sequences
3 pages
Grade 10 Math Weekly Lesson Log
No ratings yet
Grade 10 Math Weekly Lesson Log
3 pages
Math10 Summative Testq1 Upload
100% (1)
Math10 Summative Testq1 Upload
5 pages
Sequences and Series
100% (1)
Sequences and Series
12 pages
Cape Pure Math Unit 1 Series
No ratings yet
Cape Pure Math Unit 1 Series
7 pages
Mathematics: 1St Generation Modules - Version 2.0
No ratings yet
Mathematics: 1St Generation Modules - Version 2.0
18 pages
Further Maths ss1 Questions
No ratings yet
Further Maths ss1 Questions
9 pages
Math10 Quarter1 Module 3 Melc 6
No ratings yet
Math10 Quarter1 Module 3 Melc 6
12 pages

Series and Proofs Questions 0525

Uploaded by

Series and Proofs Questions 0525

Uploaded by

D

Find the values of a, b and n. [3 marks]

(b) The third term of an arithmetic sequence is 7. Find S 5 [3 marks]

(b) Find the 10th term of the sequence. [2 marks]

(b) Consider now

(b) Confirm that the statement is true for n = 3 . [2]

(a) Expand ( 2 + 5 ) 2 . [1]

(b) Prove by mathematical induction that ( 2 + 5 ) 2 n has the form

a + b 10 , where a is an odd integer and b is an even integer.

(c) Hence, show that ( 2 + 5 ) 2 n +1 has the form

c 2 + d 5 , where c, d ∈ Z for any n ∈ Z +

(b) Confirm that the statement is true for n = 3 . [2]

(a) Show that the claim is true for n  4 [2]

(a) log 3 ( x − 3) = 1 + log 9 ( x + 1) (b) log x 2 − 3 log 2 x = 2 [6+5 marks]

Solve the simultaneous equations:

(a) Prove by mathematical induction that 2 3n − 3 n is divisible by 5, for any n ≥ 1 . [7 marks]

You might also like