Question 0409

Arithmetic and Geometric Progressions (APs, GPs)

2009 Paper 1 Question 8 Variant

Question

Two sets of toys,

{A}

and

{B,}

consist of wooden blocks of decreasing lengths.

(i)

The first block of toy set

{A}

has length

{27 \textrm{ cm}}

and the lengths of the blocks form a geometric progression. The

{25 \textrm{th}}

block has length

{3 \textrm{ cm}.}

Show that the total length of all the blocks must be less than

{309 \textrm{cm,}}

no matter how many blocks there are.

[4]

Toy set

{B}

consists of only

{25}

blocks which are identical to the first

{25}

blocks of toy set

{A.}

(ii)

Find the total length,

{L \textrm{cm},}

of all the blocks of toy set

{B}

and the length of the

{13 \textrm{rd}}

block.

[3]

(iii)

Unfortunately, the manufacturer misunderstands the instructions and constructs toy set

{B}

wrongly, so that the lengths of the blocks are in arithmetic progression with common difference

{d \textrm{ cm}.}

If the total length of the

{25}

blocks is still

{L \textrm{ cm}}

and the length of the

{25 \textrm{th}}

block is still

{3 \textrm{ cm,}}

find the value of

{d}

and the length of the longest block.

[4]