Problem 76 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a_{1}=0\) and \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\) are real numbers such that \(\left|a_{i}\right|=\left|a_{i-1}+1\right|\) for all \(i\) then the A.M. of the numbers \(a_{1}, a_{2}, \ldots, a_{n}\) has value \(x\) where (A) \(x \leq-\frac{1}{2}\) (B) \(x \geq-\frac{1}{2}\) (C) \(x<-\frac{1}{2}\) (D) None of these

Step-by-Step Solution

Answer

The arithmetic mean is less than or equal to \(-\frac{1}{2}\), so the answer is (A).

Step 1: Understanding the Recurrence Relation

Given that \(|a_i| = |a_{i-1} + 1|\), this means that the absolute value of each term depends on the previous term plus one. The first term is given as \(a_1 = 0\). We need to explore how this condition impacts subsequent terms: \(|a_2| = |0 + 1| = 1\).

Step 2: Solving for Subsequent Terms

For \(a_2\), either \(a_2 = 1\) or \(a_2 = -1\). Let’s choose \(a_2 = 1\), then for \(a_3\): \(|a_3| = |1 + 1| = 2\), so \(a_3 = 2\) or \(a_3 = -2\). We notice a pattern of absolute value change in increasing positive or negative increments.

Step 3: Averaging Over n Terms

Once we choose a particular sequence for half of all terms such as \(a_2=a_4=a_6=\cdots=1, a_3=a_5=a_7=\cdots=2\), then the other half can alternatively be negative. Thus, effectively each two terms in the sequence cancel out the addition by one. This gives an average \(a_n = 0\) for large n.

Step 4: Calculate the Arithmetic Mean (AM)

Since the sequence pattern includes zero across integers, the arithmetic mean (AM) of \(a_1, a_2, \ldots, a_n\) tends towards the negative endpoint of this oscillation: \(x = -\frac{1}{2}\). Consider an average effect of alternating sequences leading to overall \(x < -\frac{1}{2}\) as -1 can dominate in one sequence.

Step 5: Determine the Correct Inequality

As observed, the average \(x\) will be either \(-\frac{1}{2}\) or less due to sign-flipping across terms without a positive trend, confirming \(x \leq -\frac{1}{2}\). Therefore, choice A is correct.

Key Concepts

Recurrence Relation

Recurrence relations are equations that recursively define sequences. Each term of the sequence is expressed as a function of its preceding terms. In our exercise, the recurrence relation provided is \(|a_i| = |a_{i-1} + 1|\). This tells us that the absolute value of any term, \(a_i\), is determined by adding one to the absolute value of the previous term, \(a_{i-1}\). The first element, \(a_1\), is zero: a known boundary condition. Hence, for \(a_2\), we calculate the absolute value from \(a_1\), giving us \(|a_2| = |0 + 1| = 1\).
This pattern allows us to systematically determine each subsequent term based on the previous term, considering positive and negative possibilities for each absolute value. Such a recurrence relation forms the backbone of analyzing the progression of the sequence in this exercise.

Absolute Value

The absolute value of a real number represents its distance from zero on the number line, denoted as \(|x|\), and is always non-negative. In a sequence, absolute value impacts how we measure each term relative to zero. Here, \(|a_i| = |a_{i-1} + 1|\) plays a crucial role.

For example, suppose we have \(a_1 = 0\). Its absolute value is 0 since distance from zero is zero. For the next term, \(a_2\), we see that \(|a_2| = |a_1 + 1| = |1| = 1\), which translates to \(a_2\) being either 1 or -1, because both give an absolute value of 1. The presence of absolute values forces us to consider both positive and negative scenarios at each step, leading to various sequence configurations based on initial terms.

Inequality

Inequalities describe the relative size or order of two values, using symbols like \(\leq\), \(\geq\), \(<\), or \(>\). In this problem, we need to determine how terms like \(x \leq -\frac{1}{2}\) relate to the sequence's arithmetic mean. This refers specifically to the result that, amid fluctuating sequence values, the average value of the terms must not exceed \(-\frac{1}{2}\).

• As terms oscillate, the averaging process converges.
• The negative terms often outweigh positive ones, leading to a consistent bias below zero.

The inequalities express our final conclusion that the arithmetic mean skewers on being negative, sealing our solution that \(x\) meets the \(\leq -\frac{1}{2}\) condition confidently.

Sequences and Series

Sequences are ordered lists of numbers following a specific rule, while series are sums of sequence terms. They can follow patterns or be defined by specific relations. In this exercise, the sequence has been influenced by its recurrence relation and use of absolute values, alternating in sign every few terms.

Observing patterns within this sequence, we notice a zigzag effect, where terms switch between positive and negative repetitively. Any given configuration of sequence terms ultimately yields a symmetry: even quantities of terms equidistant from zero, whereby their effects leverage towards a balance that averages out across the series.

The series behavior helps us conclude that the average converges near zero, sprinkling in fluctuations that affirm the condition of \(x \leq -\frac{1}{2}\). In essence, understanding series behavior in sequences guides us to visualize complex math in manageable chunks of predictability.