Thinking@TheCulture is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking Math@TheCulture is curated by KS. More of him can be found here.

This is a question from 1993 Paper 1.

The positive integers, starting at 1, are grouped into sets containing integers, as indicated below, so that the number of integers in each set after the first is twice the number of integers in the previous set.

(i) Write down the expressions, in terms of for

(a) the number of integers in the set,

(b) the first integer in the set,

(c) the last integer in the set.

(ii) Given that the integer occurs in the set, find the integer value of .

(iii) The sum of all the integers in the set is denoted by , and the sum of all the integers in all of the first sets is denoted by . Show that may be expressed as .

Hence, evaluate , correct to 4 decimal places.