Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(A\) denotes the property that two elements of \(A=\\{1\), \(5,9,13 \ldots, 1093\\}\) add up to 1094 , then the maximum number of elements in \(A\) can be (A) 126 (B) 136 (C) 137 (D) 138

Step-by-Step Solution

Answer

The maximum number of elements in set \( A \) is 137.

Step 1: Identify Elements of Set A

The set \( A \) is defined by the arithmetic sequence \( 1, 5, 9, 13, \, \ldots, 1093 \). The first term \( a_1 \) is 1, and the common difference \( d \) is 4. We need to determine all elements in this sequence that are less than or equal to 1093.

Step 2: Determine Number of Elements in Set A

The general formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \). Substituting the given values, we have the equation \( 1093 = 1 + (n-1) \cdot 4 \). Solving for \( n \) gives \( n = \frac{1092}{4} + 1 = 274 \). Therefore, there are 274 elements in set \( A \).

Step 3: Define the Condition for Pairs Adding to 1094

For a pair of elements \( (x, y) \) to sum to 1094, it must be true that \( x + y = 1094 \). In terms of set \( A \), for the smallest element \( x = 1 \), the largest possible \( y \) is 1093.

Step 4: Find Complementary Pairs

Each element \( x \) from the sequence will pair with another element \( 1094 - x \) if \( 1094 - x \) also belongs to the sequence and is distinct from \( x \). For example, \( x = 1 \) will pair with \( y = 1093 \), \( x = 5 \) will pair with \( y = 1089 \), and so on. Continue this pairing until \( x \) reaches an element just less than half of 1094.

Step 5: Determine Maximum Number of Elements

The sequence can be divided into distinct pairs \( (x, 1094-x) \). Since each pair consists of two elements, the maximum possible number of unpaired elements is one, if there is an odd number of total pairs. The largest pairable group is \( \frac{274}{2} = 137 \) distinct pairs, leaving 0 unpaired elements.

Key Concepts

Pairing Conditions in an Arithmetic Sequence

When dealing with arithmetic sequences, pairing refers to the process of matching elements that satisfy a certain condition. In the given exercise, the condition is that any two elements of set \( A \) must add up to 1094. This means for each element \( x \) in the sequence, we can look for a corresponding \( y \) such that \( x + y = 1094 \).
\(A\) is formulated as a sequence starting at 1 with a common difference of 4, making it an arithmetic sequence.
Understanding the pairing condition is crucial. By determining that \( y = 1094 - x \), you can find out if \( y \) also belongs in the sequence by checking whether it conforms to the sequence's rule. If both \( x \) and \( y \) are in the sequence and differ from each other, then they form a valid pair.

• Each element pairs with another unique element to add up to 1094.
• Pairs are only formed if both numbers are distinct and present in the sequence.

This helps break down the sequence into couplets of two that can simplify calculations and provide insights into the structure of the sequence.

Calculating the Maximum Number of Elements

The key to finding the maximum number of elements under the given condition lies in the arithmetic sequence properties.
Set \( A \) includes numbers from 1 to 1093 with a step of 4, meaning there are 274 elements in total as calculated using the nth term formula \( a_n = a_1 + (n-1)d \).
The maximum number of elements that can populate the sequence while maintaining the sum-to-1094 condition depends on how many distinct pairs can be formed.

Each pair consists of two numbers, and by calculating \( \frac{274}{2} = 137 \), it becomes clear that 137 pairs or couplets can be formed, utilizing all the elements in the sequence.

• Every element potentially has a partner if all conditions are satisfied.
• The sequence must have an even number of elements to form complete pairings without leftovers.

The understanding of maximum elements guides how sequences can be paired responsibly based on their array count.

Understanding the Arithmetic Progression

An arithmetic progression (AP) is a sequence where each term after the first is derived by adding a constant difference to its predecessor. In the context of this problem, the sequence starts at 1 and advances by a common difference of 4, progressing as 1, 5, 9, 13, ..., up to 1093.

To understand and manage an AP, particularly when dealing with so many terms, it's helpful to express the elements in terms of the first term \( a_1 \) and the common difference \( d \). The general nth term equation \( a_n = a_1 + (n-1)d \) simplifies locating any element and calculating total elements as demonstrated for this sequence.
Knowing how to work with APs allows one to easily establish more complex relationships as seen with pairing.

• Arithmetic sequences mean predictable increments from term to term.
• AP properties ease calculations of terms' positions and relationships.

This structured information helps simplify and better organize computations involving progression and element pairings.