Question 0413

Arithmetic and Geometric Progressions (APs, GPs)

2013 Paper 1 Question 7 Variant

Question

Charlie is cutting off pieces of rope from a long roll of rope. The first piece he cuts off is

{128}

cm long and each successive piece is

{\frac{2}{3}}

as long as the preceding piece.

(i)

The length of the

{n\textrm{th}}

piece of the rope is

{p}

cm. Show that

\ln p = (An +B) \ln 2 + (Cn+D) \ln 3,

for constants

{A,B,C}

and

{D}

to be determined.

[3]

(ii)

Show that the total length of string cut off can never be greater than

{384}

cm.

[2]

(iii)

How many pieces must be cut off before the total length cut off is greater than

{380}

cm?

[4]