A factory’s production yield decreases in a predictable pattern: the quantity…

MATHEMATICS · Multi-part · A Level · Mathematics · Sequences and series · Sequences and series

Problem

A factory’s production yield decreases in a predictable pattern: the quantity produced during cycle

n

a r^{n}

units, where

n=0

corresponds to the first cycle. Quality-control data show that the total production over all cycles must be

1{,}200

units, while the sum of the first four cycles’ production is

800

units. (a) Write two separate equations that express the given total production constraints in terms of

a

and

r

. Explain the origin of each equation. (b) Solve the system of equations from part (a) to determine the exact values of

a

and

r

. (c) Demonstrate explicitly that the value of

r

obtained in part (b) satisfies the condition necessary for the relevant infinite series to converge. (d) Interpret the numerical value of

r

in the context of the factory’s production process, including what it implies about the relative sizes of successive production cycles.