[email protected] is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.

Thinking [email protected] is curated by KS. More of him can be foundĀ here.

This is a question from 1993 Paper 1.

The positive integers, starting at 1, are grouped into sets containing 1, 2, 4, 8, \ldots 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.

\{ 1 \}, \{ 2, 3 \}, \{ 4, 5, 6, 7 \}, \{ 8, 9, 10, 11, 12, 13, 14, 15 \}, \ldots

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

(a) the number of integers in the r^{th} set,

(b) the first integer in the r^{th} set,

(c) the last integer in the r^{th} set.

(ii) Given that the integer 1,000,000 occurs in the r^{th} set, find the integer value of r.

(iii) The sum of all the integers in the 20^{th} set is denoted by S, and the sum of all the integers in all of the first 20 sets is denoted by T. Show that S may be expressed as 2^{18}(3 \times 2^{19} - 1).

Hence, evaluate \frac{T}{S}, correct to 4 decimal places.


Leave a Comment

three × three =

Contact Us

CONTACT US We would love to hear from you. Contact us, or simply hit our personal page for more contact information

Not readable? Change text. captcha txt

Start typing and press Enter to search