Problem 81 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If for a real number \(x,[x]\) denotes the greatest integer less than or equal to \(x\), then for any \(n \in N\) $$ \left[\frac{n+1}{2}\right]+\left[\frac{n+2}{4}\right]+\left[\frac{n+4}{8}\right]+\left[\frac{n+8}{16}\right]+\ldots= $$ (A) \(n\) $$ \begin{aligned} &\text { (B) } n-1 \\ &\text { (D) } n+2 \end{aligned} $$ (C) \(n+1\)

Step-by-Step Solution

Answer

The sum is \(n\) for any \(n \in N\). Answer: (A) n.

Step 1: Understand the function [x]

The bracket notation \([x]\) signifies the greatest integer less than or equal to \(x\). For example, \([3.7] = 3\) and \([4] = 4\).

Step 2: Analyze terms and pattern formation

Examine the terms \(\left[\frac{n+1}{2}\right], \left[\frac{n+2}{4}\right],\) and so on. These are of the form \(\left[\frac{n+k}{2^k}\right]\). The pattern shows that each consecutive term reduces the power of 2 in the denominator while increasing the numerator by the additional constant integer value.

Step 3: Consider small values of n to predict the pattern

Plug in small values of \(n\) such as \(n = 1, 2, 3\) and manually compute the values of the sequence to determine what the sum approaches.

Step 4: Calculate for n = 1

For \(n=1\), calculate \(\left[\frac{2}{2}\right] + \left[\frac{3}{4}\right] + \left[\frac{5}{8}\right] + \ldots = 1 + 0 + 0 + \ldots = 1\). The sum is \(1\) which is \(n\).

Step 5: Calculate for larger n values

Using the same procedure for \(n=2\), and \(n=3\):\[n=2: \left[\frac{3}{2}\right] + \left[\frac{4}{4}\right] + \left[\frac{6}{8}\right] + \ldots = 1 + 1 + 0 + \ldots = 2\] and \[n=3: \left[\frac{4}{2}\right] + \left[\frac{5}{4}\right] + \left[\frac{7}{8}\right] + \ldots = 2 + 1 + 0 + \ldots = 3\].

Step 6: Establish general pattern and conclusion

The observed sums \(1, 2, 3\) for \(n = 1, 2, 3\) respectively aligns directly with \(n\). The pattern indicates that the sum of the series equals \(n\).

Step 7: Validate the pattern

The series formed is essentially adding ceiling elements consecutively decreasing in their contribution as \(k\) increases further. This collectively sums exactly to \(n\) for all integers \(n\).

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([x]\), is a mathematical function that assigns to any real number \(x\) the greatest integer less than or equal to \(x\). It's particularly useful when dealing with scenarios that require rounding down a number to the nearest whole number. For example, when \(x = 5.8\), the greatest integer function gives \([5.8] = 5\). Importantly, if \(x\) is already an integer, such as 5, then \([5] = 5\) as well. This function is sometimes also referred to as the "floor" function and creates a step-like pattern on a graph as it jumps from one integer to the next.When solving problems involving the greatest integer function, it helps to think about what the largest whole number is just before reaching \(x\). In the case of fractions and whole numbers that result from division, this concept can be crucial in understanding how to simplify complex expressions that involve parts of integers.

Sequence Pattern

Sequence patterns are all about recognizing recurring themes or rules in a sequence of numbers, which follow specific arrangements derived from a formula or a list. In the problem at hand, we're dealing with terms like \(\left[\frac{n+k}{2^k}\right]\), where each term follows a specific pattern. Here, each term involves increasing the numerator by a steady increment \(k\), while the denominator grows exponentially as a power of 2. Understanding this sequence pattern is key to decoding the behavior of these terms as \(n\) increases, and it helps predict the outcome of the series. By observing small cases (like \(n = 1, 2, 3\)), students can better comprehend how these terms behave and form predictions for larger values. Recognizing these patterns helps simplify calculations and analysis of complex sequences.

Summation

Summation is the process of adding a sequence of numbers, which allows us to find their total or sum. It's a fundamental operation in mathematics and is often denoted using the big sigma notation \(\Sigma\). For the series in our problem, summation involves adding all the terms of the form \(\left[\frac{n+k}{2^k}\right]\). By evaluating these terms individually and summing them up, we can see the total result for a given \(n\). Often, performing these calculations with specific small numbers like \(n=1, 2, 3\) helps students visualize the overall behavior. These initial calculations can reveal patterns or insights that directly lead to understanding how the whole series or sequence sums up to something predictable, like the integer \(n\) in this example. Doing manual calculations for specific values can build intuition and solidify understanding of the series' summation behavior.

Mathematical Series

A mathematical series is essentially a sequence of numbers that are added together to form a sum. Series often arise in the study of sequences, especially when there's interest in the total of all terms in a sequence. In tackling the given exercise, the series takes the form \(\left[\frac{n+1}{2}\right] + \left[\frac{n+2}{4}\right] + \left[\frac{n+4}{8}\right]\), continuing with denominators increasing as powers of 2. Each term presents a reduced impact as \(k\) increases, which shapes the overall sum's behavior. By looking at specific series values for small integers, such as 1, 2, and 3, we can derive general rules or formulas for predicting the behavior of larger series. This specific problem is an example of how many mathematical series converge to a simple result, like \(n\), through the strategic structure of the terms and recognizing the diminishing returns of each additional term.