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Chapter 1: Problem 71

For each positive integer \(n\), let $$ N(n)=\left\lceil\frac{n}{2}\right\rceil+\left\lceil\frac{n}{4}\right\rceil+\left\lceil\frac{n}{8}\right\rceil+\cdots+\left\lceil\frac{n}{2^k}\right\rceil, $$ where \(k\) is the unique integer such that \(2^{k-1} \leq n<2^k\), and \(\lceil x\rceil\) denotes the smallest integer greater than or equal to \(x\). For which numbers \(n\) is \(N(n)=n ?\)

Short Answer

Expert verified
The numbers \(n\) for which \(N(n) = n\) are exactly those of the form \(2^m - 1\) for any positive integer \(m\).

Step by step solution

01

Understanding the given function

The function is defined as \(N(n)=\lceil\frac{n}{2}\rceil + \lceil\frac{n}{4}\rceil + \lceil\frac{n}{8}\rceil + \cdots + \lceil\frac{n}{2^k}\rceil\), where \(k\) is such that \(2^{k-1} \leq n < 2^k\).
02

Determine the value of \(k\)

Find the integer \(k\) such that \(2^{k-1} \leq n < 2^k\). This means \(k = \lceil \log_2(n+1) \rceil\), since \(2^k\) is the smallest power of 2 greater than or equal to \(n+1\).
03

Evaluate the function \(N(n)\)

Calculate the sum \(N(n) = \lceil\frac{n}{2}\rceil + \lceil\frac{n}{4}\rceil + \lceil\frac{n}{8}\rceil + \cdots + \lceil\frac{n}{2^k}\rceil\). Notice that as the series progresses, the terms \(\lceil\frac{n}{2^i}\rceil\) become 1 once \(2^i > n\). Hence, from a certain point, the terms simply amount to 1.
04

Check for equality in each term

For \(N(n) = n\) to hold, the sum of ceil values should precisely add up to \(n\). Observing the terms, when \(n = 2^m - 1\) for any integer \(m\), the ceil values are distributed as required under \(\left\lceil\frac{n}{2}\right\rceil + \left\lceil\frac{n}{4}\right\rceil + \left\lceil\frac{n}{8}\right\rceil + \cdots + \left\lceil1\right\rceil\). Thus, for each such \(m\), i.e., \(n = 2^m - 1\), all values in the series terminate to 1 accurately.
05

Summarize the final solution

The positive integers \(n\) that satisfy \(N(n) = n\) are the ones of the form \(2^m - 1\) for \(m\) being a positive integer. Therefore, the required values are 1, 3, 7, 15, 31, etc.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ceil function
The ceil function, denoted as \(\lceil x \rceil\), rounds a number up to the nearest integer. For example:

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