Problem 123 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are three natural numbers in A.P. such that \(a+b+c=21\), then the possible number of values of \(a, b, c\) is (A) 13 (B) 14 (C) 15 (D) 16

Step-by-Step Solution

Answer

The possible number of values is 6.

Step 1: Understanding Arithmetic Progression

In an arithmetic progression (A.P.), there is a common difference between consecutive terms. If \(a, b, c\) are in A.P., then \(b - a = c - b\), which implies \(b = \frac{a+c}{2}\).

Step 2: Write Equation for Sum

Given that \(a + b + c = 21\), and knowing \(b = \frac{a+c}{2}\), substitute \(b\) in the equation: \(a + \frac{a+c}{2} + c = 21\).

Step 3: Simplify and Solve for Common Difference

Multiply the whole equation by 2 to eliminate the fraction: \(2a + a + c + 2c = 42\), which becomes \(3a + 3c = 42\). Simplifying gives \(a + c = 14\). Therefore, \(b = \frac{14}{2} = 7\).

Step 4: Find Possible Values for a, c

Since \(a, c\) are natural numbers with \(a + c = 14\), and \(a < b < c\) or \(a < 7 < c\), find valid \(a, c\) pairs with \(b = 7\).

Step 5: List and Count Possible Combinations

Considering \(a + c = 14\) and valid conditions for \(a\) (\(a < 7\)), the possible pairs \((a, c)\) are \((1, 13)\), \((2, 12)\), \((3, 11)\), \((4, 10)\), \((5, 9)\), \((6, 8)\). There are 6 combinations, satisfying \(a < b < c\).

Step 6: Conclusion on Number of Values

Considering symmetry, \(a, b, c\) can also obey \(a = b = c\), which occurs when each equals 7. Therefore, there's no distinct pair here, and it accounts only if \(a < 7\). The total therefore is 6 pairs listed earlier.

Key Concepts

Natural Numbers

Natural numbers are a fundamental part of our number system. They are the basic counting numbers starting from 1 and moving upwards without any decimal or fractional part.
This sequence of numbers includes 1, 2, 3, 4, and so on.Natural numbers are often used in arithmetic progressions because they provide a straightforward way to consider patterns and sequences. In an arithmetic progression, terms usually follow a specific formula or pattern, making it easier to predict future terms. Additionally, because natural numbers are easy to understand and manipulate, they form a good base for learning about more complex mathematical concepts.
In the context of our problem, the terms of the sequence \(a, b, c\) are natural numbers, illustrating a simple but powerful example of using these basic counting numbers in more complex mathematical situations.

Sum of Sequences

A sequence is an ordered list of numbers. The sum of a sequence is the result of adding every number in that list.
This concept is fundamental in many mathematical applications.When we're working with arithmetic progressions, the sum of the sequence is especially important because it can help us solve problems and find unknown terms.
In this particular exercise, knowing the total sum of the sequence members allows us to apply other mathematical tools, like substitution and simplification, to determine the possible values for terms in the sequence.For the sequence with terms \(a, b, c\) in the problem, the sum is given as 21. Using this information, combined with the property that these numbers are in arithmetic progression, allows us to set up equations and rearrange them to find all possible sets of values.

Consecutive Terms in Arithmetic Progressions

In arithmetic progressions, consecutive terms increase by the same constant difference. This property is what defines the progression and allows us to use very specific mathematical techniques to explore and solve problems linked to them.
For three numbers \(a, b, c\) forming an arithmetic progression, the middle term \(b\) can be calculated using the formula: \(b = \frac{a+c}{2}\).Understanding consecutive terms helps tremendously when solving arithmetic progression problems. In our problem setup, it's important to realize that if \(a < b < c\), then the difference between each consecutive pair of numbers remains constant. This simple yet powerful method of defining relationships between terms allows us to manipulate algebraic expressions easily and calculate potential values for each of the terms within the constraints of the problem...Knowing how consecutive terms relate also aids in finding the range of possibilities for numbers in sequences, understanding that each consecutive pair \(a, b\) and \(b, c\) needs to maintain the pattern we've established with our arithmetic progression rules.