1. Calculator Question (4 marks) - Ascientia

Mathematics AA | IB Style Questions

Number and Algebra: Sequences and Series

1.

 n arithmetic sequence has 7 and 15 as its first two terms respectively.

A

Calculator Question (4 marks)

(a) Write down, in terms of n, an expression for the nth term, an. [2]

a1 = 7 and d = 8

an = a1 + (n �C 1)d

an = 8n �C 1 [2]

(b) Find the number of terms of the sequence which are less than 500. [2]

8n �C 1 < 500

8n < 501

n < 62.625

2.

Therefore, there are 62 terms less than 500. [2]

The first term of a geometric sequence, u1, is 1.5 and the common ratio, r, is 1.5.

Calculator Question (5 marks)

(a) Write down, in terms of n, an expression for the nth term, un. [2]

u1 = 1.5 and r = 1.5

un = u1rn-1

un=1.5(1.5)n-1=1.5n

(b) What is the largest value of n for which term, un is less than 40. [3]

n <

ln(40)

ln(1.5)

n < 9.09

3.

Therefore, the largest possible value of n is 9.

An arithmetic series is given by: -4 + 3 + 10 + 17 + ...

 ind the least number of terms so that the sum of the series is greater than 10 000.

F

Calculator Question (6 marks)

Using Sn=

Sn=

n

2

n

2

( 2u1 + (n �C 1)d) with u1 = -4 and d = 7 [M1][A1]

( 7n �C 15) [A1]

Solving Sn > 10000 for n �� (7n2 �C 15n �C 20000 > 0) [M1]

n > 54.53�� [A1]

The least number of terms is 55. [A1][N4]

Worksheet 1911

? 2019 Ascientia Ltd.

Mathematics AA | IB Style Questions

Number and Algebra: Sequences and Series

4.

 circular disc is cut into 10 sectors (wedges) whose areas are in an arithmetic sequence.

A

The angle of the largest sector is three times the angle of the smallest sector. Find the size

of the angle of the smallest sector.

Non-Calculator Question (6 marks)



The area of the sectors follow an arithmetic series. The area of a sector is given

1

by A = 2 r2��.

Since the radius across all sectors remains constant, the angles of the sectors

also follow the same arithmetic sequence as the areas. Consequently, we can

say that

u1 = a and u10 = 3a, where a is the angle of the smallest sector.

Since the sectors form a circle;

Sn = 2�� (or 360��)

n

2

Sn=

10

(u1+un)

2 (a + 3a) = 2�� (or 360��)

a =

5.

��

10

(or 18��)

 he first and fifth terms of geometric series with common ratio, r > 0, are 15 and

T

respectively.

Non-Calculator Question (Total 6 marks)

5

27

Find

(a) an expression for the sum of the first n terms; [4]

u1 = 15 , u5 =

u5 = u1r4 =

r4 =

r =

5

27

1

81

1

since

3

5

27

r>0

Therefore Sn =

u1(1 �C rn)

1�Cr

=

45

2

(1-

( 13 ) ) or -452 (( 13 ) �C r)

n

n

[4]

(b) the sum to infinity of the series. [2]

S�� =13

15

1�C

u1

1�Cr

= 45

2

=

[2]

Worksheet 1911

? 2019 Ascientia Ltd.

Mathematics AA | IB Style Questions

Number and Algebra: Sequences and Series

6.

 he third term of an arithmetic sequence is 14 and the sum of the first 6 terms is 99. Find

T

the first term and common difference of this arithmetic sequence.

Non-Calculator Question (Total 6 marks)

Using u3 = u1 + 2d

? u1 + 2d = 14

Using S6 =

6

2

(2u1 + 5d)

? 3(2u1 + 5d) = 99

Attempting to solve simultaneously

u1 = 4, d = 5

11

7. 

The ratio of the 7th term to the 10th term of an arithmetic progression is . Given that the

14

product of the 1st and 4th term of this progression is 160, find the possible values for the first

term and common difference of this sequence.

Non-Calculator Question (Total 7 marks)

Let the arithmetic sequence be written as a, a + d, a + 2d, ...

Then

(a + 6d)

(a + 9d)

=

11

14

14a + 84d = 11a + 99d

a = 5d

Using u1 �� u4 = 160

�� a(a + 3d) = 160

5d(5d + 3d) = 160

d = ��2 and a = ��10

8.

A sum of $200 is invested. Calculator Question (Total 7 marks)

(a) If the interest is compounded annually at a rate of 5% per year, find the total value FV of

the investment after 25 years. [3]

FV = 200(1 + 0.05)20

FV = $677 (accept $677.27)

5

(b) If the interest is compounded monthly at a rate of 12 % per month, find the minimum

number of months for the value of the investment to exceed FV. [4]

5 n

)

1200

5

ln(1+ 1200 ) >

100(1 +

> 677

n

ln(3.386��)

n > 293.35

Therefore minimum number of months required is n = 294

This is equivalent to

294

12

= 24.5years.

Worksheet 1911

? 2019 Ascientia Ltd.

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