Question 0409

Arithmetic and Geometric Progressions (APs, GPs)
2009 Paper 1 Question 8 Variant

Question

Two sets of toys, A{A} and B,{B,} consist of wooden blocks of decreasing lengths.
(i)
The first block of toy set A{A} has length 75 cm{75 \textrm{ cm}} and the lengths of the blocks form a geometric progression. The 21st{21 \textrm{st}} block has length 3 cm.{3 \textrm{ cm}.}
Show that the total length of all the blocks must be less than 505cm,{505 \textrm{cm,}} no matter how many blocks there are.
[4]
Toy set B{B} consists of only 21{21} blocks which are identical to the first 21{21} blocks of toy set A.{A.}
(ii)
Find the total length, Lcm,{L \textrm{cm},} of all the blocks of toy set B{B} and the length of the 11st{11 \textrm{st}} block.
[3]
(iii)
Unfortunately, the manufacturer misunderstands the instructions and constructs toy set B{B} wrongly, so that the lengths of the blocks are in arithmetic progression with common difference d cm.{d \textrm{ cm}.}
If the total length of the 21{21} blocks is still L cm{L \textrm{ cm}} and the length of the 21st{21 \textrm{st}} block is still 3 cm,{3 \textrm{ cm,}} find the value of d{d} and the length of the longest block.
[4]

Answer