Analysis and Approaches for IBDP Maths Ebook 1 | Page 71

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3. The first three terms of an infinite geometric sequence are
where m� .
log 2
m 1
x � ,
2
log2
x ,
log 2
m 2
x � ,
(a) (i) Write down an expression for the common ratio, r .
(ii)
Hence, show that m satisfies the equation
(b) (i) Find the two possible values of m .
m
2
� m�6� 0.
[4]
(ii) Find the possible values of r .
(c) Consider the value of r such that �1� r � 1. Find S �
.
(d) Consider the value of r such that r �� 1. Find S
6
when
1
x � .
2
4. The first four terms of an infinite geometric sequence are u
1
, u2 �m� 2 , u3 � 9 ,
m �17
u4
� , where m� .
4
(a) (i) Write down an expression for the common ratio, r .
[6]
[2]
[4]
6
(ii) Hence, show that m � 10 or m �� 29 .
[5]
(b) Find the two possible values of r .
[3]
(c) Consider a sequence of squares with lengths u
1
, u
2
, u
3
, …, that is, their lengths
form a geometric sequence. This sequence has a finite sum.
(i)
(ii)
State which value of r leads to this sum and justify your answer.
The squares are created indefinitely. Find the total sum of the areas of all
the squares.
[7]
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