Cambridge International Mathematics

Investigation and modelling questions (Chapter 34) 1

21

a Show that

1

n

¡

1

n+1

=

1

n(n+1)

.

n

1

n

¡

1

n+1

1

n(n+1)

1

1

1

¡

1

2

1

1 £ 2

2 ......¡...... ......

3 ......¡...... ......

4 ......¡...... ......

99 ......¡...... ......

100 ......¡......

1

100 £ 101

b Copy the following table, completing the rows for n=2,

3 , 4 , 99 and 100.

c Useaand your table to find another expression for

1

1 £ 2

+

1

2 £ 3

+

1

3 £ 4

+::::::+

1

100 £ 101

:

Write your answer as a single fraction.

22

The height of a cylinder is 10 cm and its radius is 2 : 5 cm.

a

A piece of string is wound evenly once around the curved surface of the cylinder, starting at a

point A on the circumference of the top circular face and finishing at B, vertically below A.

A sketch of the net of the curved surface of the cylinder, together with the string AB, is shown

above in the diagram on the right. Calculate the length of the string, AB.

b

Another string, starting at A, is wound evenlytwicearound the cylinder, finishing again at B.

Calculate the length of this string.

c Sketch the net when a string, starting at A, is wound evenly three times around the cylinder,

finishing at B.

d A string is wound evenlyntimes around the cylinder, from A to B. Find a formula, in terms of

n, for the length of the string.

A

B

2.5 cm

10 cm

A

B

A

B

A

B

Adapted from Nov 1993, Paper 4

Adapted from Nov 1999, Paper 4

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