Section 3: Sequences and series

Name the first term and the common difference or ratio, then write the general term before anything else.

Section A: multiple choice

Question 1Section A: multiple choice

In an arithmetic progression

a_1, a_2, a_3, \ldots

a_5+a_9+a_{13}

=69

and

a_3a_{15}

=205

. The greatest possible value of the sum of the first

20

terms is

Question 2Section A: multiple choice

The sum of the first

n

terms of a sequence is

S_n

=3n^2+5n

for every positive integer

n

. The 15th term of the sequence is

Question 3Section A: multiple choice

For two arithmetic progressions, the ratio of the sums of their first

n

terms is

(5n+3):(3n+11)

for every positive integer

n

. The ratio of their 9th terms is

Question 4Section A: multiple choice

A geometric progression of real terms has third term

12

and sixth term

96

. The sum of its first

8

terms is

Question 5Section A: multiple choice

The arithmetic progression

4, 9, 14, \ldots

has

100

terms and the arithmetic progression

3, 10, 17, \ldots

has

80

terms. How many numbers appear in both progressions?

Question 6Section A: multiple choice

Three real numbers in geometric progression have product

216

and sum

19

. The sum of their squares is

Question 7Section A: multiple choice

For the arithmetic progression

47, 43, 39, \ldots

, let

S_n

be the sum of the first

n

terms. The greatest value of

S_n

Question 8Section A: multiple choice

In a geometric progression of positive terms, every term equals the sum of the two terms that immediately follow it. The common ratio is

Question 9Section A: multiple choice

The interior angles of a convex polygon, in increasing order, form an arithmetic progression with smallest angle

100^\circ

and common difference

10^\circ

. Every interior angle of a convex polygon is less than

180^\circ

. The number of sides of the polygon is

Question 10Section A: multiple choice

An infinite geometric progression of real terms has sum

9

, and the sum of its first two terms is

5

. The sum of all possible values of its first term is

Question 11Section A: multiple choice

The 3rd, 7th and 15th terms of an arithmetic progression with non-zero common difference are, in that order, the first three terms of a geometric progression. The sum of the first

10

terms of the arithmetic progression is

130

. The sum of the first

6

terms of the geometric progression is

Question 12Section A: multiple choice

An infinite geometric progression of real terms has sum

5

. Which of the following could be its first term?

Question 13Section A: multiple choice

The sum of the first

n

terms of a sequence

T_1, T_2, T_3, \ldots

S_n

=2^{n+1}+3n^2-2

. The value of

T_2

{}+T_4

{}+T_6

{}+\cdots

{}+T_{20}

Question 14Section A: multiple choice

An arithmetic progression has

31

terms

a_1, a_2, \ldots, a_{31}

, and

a_3

{}+a_8

{}+a_{13}

{}+a_{19}

{}+a_{24}

{}+a_{29}

=186

. The sum of all

31

terms is

Question 15Section A: multiple choice

A sequence has

a_1

=3

and

a_{n+1}

=2a_n+3

for

n

\ge 1

. The value of the sum

a_1+a_2+\cdots+a_{10}

Question 16Section A: multiple choice

Each term of an infinite geometric progression of positive terms is

4

times the sum of all the terms that follow it. The sum of the progression is

10

. Its third term is

Question 17Section A: multiple choice

Some arithmetic means are inserted between

5

and

47

, and their sum is

286

. The 4th of these means (counting from

5

) is

Question 18Section A: multiple choice

Some positive geometric means are inserted between

3

and

192

, and their sum is

186

. The product of these geometric means is

Question 19Section A: multiple choice

Two positive numbers

p

>q

are such that their arithmetic mean exceeds their geometric mean by

2

, and

p-q

=12

. The value of

pq

Question 20Section A: multiple choice

The arithmetic mean and the geometric mean of two positive numbers

p

>q

are in the ratio

5:4

. Then

p:q

Question 21Section A: multiple choice

For

x

>0

, the least value of

\frac{(x+2)(x+8)}{x}

Question 22Section A: multiple choice

Positive real numbers

x, y, z

satisfy

xyz

=64

. The least value of

x+2y+4z

Question 23Section A: multiple choice

Positive real numbers

x

and

y

satisfy

x+y

=12

. The greatest value of

x^2y

Question 24Section A: multiple choice

For

0

<\theta

<\frac{\pi}{2}

, the least value of

\sec^2\theta

{}+16\csc^2\theta

Question 25Section A: multiple choice

Between two positive numbers

p

and

q

, one arithmetic mean

A

is inserted, and separately two geometric means

G_1

and

G_2

are inserted. If

A

=15

and

G_1G_2

=144

, then

G_1^3+G_2^3

equals

Question 26Section A: multiple choice

Two positive numbers have arithmetic mean

25

. Three positive geometric means are inserted between them, and the middle one is

20

. The sum of the three geometric means is

Question 27Section A: multiple choice

Positive real numbers

x, y, z

satisfy

x+y+z

=12

. The greatest value of

xy^2z^3

Question 28Section A: multiple choice

The quadratic equation

x^2-kx+16

=0

, where

k

is real, has both roots real and positive. The least possible value of

k

Section B: numerical value

Sequences and seriesSection B: numerical value · 4 marks

Find the sum of all the integers between

100

and

600

that are divisible by

7

but not divisible by

3

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Sequences and seriesSection B: numerical value · 4 marks

An arithmetic progression

a_1, a_2, a_3, \ldots

has

a_1

=3

and common difference

4

. Find the value of

a_1^2

{}-a_2^2

{}+a_3^2

{}-a_4^2

{}+\cdots

{}+a_{19}^2

{}-a_{20}^2

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Sequences and seriesSection B: numerical value · 4 marks

An arithmetic progression with non-zero common difference and a geometric progression both have first term

3

. Their second terms are equal, and the 4th term of the arithmetic progression equals the 3rd term of the geometric progression. Find the sum of the first

10

terms of the arithmetic progression added to the sum of the first

10

terms of the geometric progression.

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Sequences and seriesSection B: numerical value · 4 marks

Find the least value of

x^3+\frac{48}{x}

for

x

>0

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Sequences and seriesSection B: numerical value · 4 marks

Positive real numbers

x

and

y

satisfy

xy^2

=108

. Find the least value of

x+y

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.

Sequences and seriesSection B: numerical value · 4 marks

Between

2

and

162

m

arithmetic means

A_1

<A_2

<\cdots

<A_m

are inserted, and separately

m

positive geometric means

G_1

<G_2

<\cdots

<G_m

whose common ratio is an integer. The sum of the arithmetic means exceeds the sum of the geometric means by

168

. Find

A_1G_1

{}+A_2G_2

{}+\cdots

{}+A_mG_m

Enter the answer as an integer.

Type one integer, such as 42 or -3. Work on paper first.