Problem 79 - Algebra for JEE ADVANCED

Question

In how many different ways can the first 12 natural numbers be divided into three different groups such that numbers in each group are in A.P.? a. 1 b. 5 c. 6 d. 4

Step-by-Step Solution

Answer

5 ways to divide the numbers into groups of A.P.

Step 1: Understanding the Problem

We need to divide the first 12 natural numbers (1 through 12) into three groups such that each group forms an arithmetic progression (A.P.). Each number must belong to one group, and no two groups should be identical. We'll aim to find all such arrangements.

Step 2: Selecting the Common Difference

For a sequence to form an A.P., a common difference (d) is required between consecutive terms. The first few natural numbers allow for several choices of "d", but we will focus on those that allow the entire set of 12 numbers to be split into three distinct A.P. groups efficiently.

Step 3: Finding Valid Groups

One efficient grouping is choosing sequences with a reasonable common difference, such as 3, which can lead to the following groups: (1, 4, 7, 10), (2, 5, 8, 11), and (3, 6, 9, 12). All these groups have the same common difference of 3, have four elements each, and do not overlap.

Step 4: Count All Possible Arrangements

Given that all numbers are used when split with a common difference correctly, these groups will be formed for each eligible arrangement of "d". Calculate or recognize patterns in how sequences line up and repeat for different values of common differences and patterns.

Key Concepts

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This difference is known as the "common difference," and it can be either positive or negative. For example, in the sequence 2, 4, 6, 8, the common difference is 2.
Each A.P. can be expressed in the form of:

• First term: \( a \)
• Common difference: \( d \)
• n-th term: \( a + (n-1)d \)

In the exercise, the task is to group numbers into A.P.s. Here, we looked for sequences that divide the first 12 natural numbers into distinct groups of four terms each. Choosing an appropriate common difference, such as 3, makes it possible to organize all numbers neatly into sequences like (1, 4, 7, 10), ensuring that each group is distinct and covers all numbers.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, excluding fractions and decimals. They are denoted by \( \{ 1, 2, 3, 4, ... \} \).
These numbers are straightforward for operations like addition and multiplication.
In the context of the problem, we specifically use the first twelve natural numbers. This set must be divided into groups that fit the A.P. criteria. Natural numbers provide a simple yet rich playground for practicing arithmetic sequences, as intuitively, they foster easy calculations and pattern recognition.

Grouping Problems

Grouping problems involve dividing a set of elements into subsets based on certain conditions or rules. In this problem, the challenge is to divide the first 12 natural numbers into three groups. Each group must create an arithmetic progression and uniquely use each number once.
Key considerations in such problems:

• The total set must cover all elements without omission.
• Each subset (group) needs to satisfy the condition (A.P. in this case).
• No two groups should be identical, ensuring unique solutions.

The sequence needs an equal distribution, which is pivotal when determining the common difference. Here, using a difference of 3 intelligently organizes the numbers into groups satisfying the conditions perfectly.

Permutations

Permutations represent various arrangements or sequences of a set of items. While permutations usually allow any order, this problem's limitation forms permutations within constraints of arithmetic progression.
In simpler terms, although each number can freely be placed initially, the restriction arises from needing to maintain an arithmetic progression. This adds layers to the permutation—where order matters alongside the necessity of forming distinct groups.
The permutations problem here subtly ensures all numbers must fit within these precise group rules. This means cycles and patterns leverage the arithmetic difference, aligning to reflect permutations that fulfill the condition while maintaining full coverage of the number set.