Investigation and

modelling questions

34

Contents:

A Investigation questions

B Modelling questions

1 It is given that 12 +2^2 +3^2 +4^2 +5^2 +::::::+n^2 =

n(n+ 1)(2n+1)

k

where k 2 Z:

a If n=1, the LHS is 12.

If n=2, the LHS is 12 +2^2.

Use n=1to find the value ofk, and check that you get the same value ofkfor n=2and

n=3.

b Use the given formula to find the value of 12 +2^2 +3^2 +4^2 +::::::+ 100^2.

c Notice that 22 +4^2 +6^2 =(2£1)^2 +(2£2)^2 +(2£3)^2

=2^212 +2^222 +2^232

=2^2 (1^2 +2^2 +3^2 )

Hence,

i findmif 22 +4^2 +6^2 +8^2 +::::::+ 100^2 =2^2 (1^2 +2^2 +3^2 +::::::+m^2 )

ii find the value of 22 +4^2 +6^2 +8^2 +::::::+ 100^2 :

d Usebandciito find the value of 12 +3^2 +5^2 +7^2 +::::::+99^2 :

e Use some of the previous answers to find the value of:

12 ¡ 22 +3^2 ¡ 42 +5^2 ¡ 62 +::::::+99^2 ¡ 1002.

A INVESTIGATION QUESTIONS

Many of the questions in and are adapted from past examination papers for IGCSE

Mathematics by permission of the University of Cambridge Local Examinations Syndicate. The

course is a different syllabus from that followed by students of the course, but has many features in

common. These questions are certainly appropriate for practising mathematical techniques and applications

relevant to the curriculum, but do not necessarily represent the style of question that will be

encountered on the examination papers. Teachers are referred to the specimen papers of the

syllabus for a more representative group of questions. The University of Cambridge Local Examinations

Syndicate bears no responsibility for the example answers to questions taken from its past question papers

which are contained in this publication.

Chapters 33 34

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