Problem 91 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c, d\) are distinct integers in A.P. such that \(d=a^{2}\) \(+b^{2}+c^{2}\), then \(a+b+c+d=\) (A) 2 (B) 1 (C) 0 (D) None of these

Step-by-Step Solution

Answer

The sum is 2.

Step 1: Define an Arithmetic Progression (A.P.)

An A.P. (Arithmetic Progression) is a sequence of numbers such that the difference of any two successive members is a constant. Let's assume the A.P. as \( a, a+k, a+2k, a+3k \), where \( k \) is the common difference.

Step 2: Write the equation for d

Given that \( d = a^2 + b^2 + c^2 \). In terms of \( a \) and \( k \), we have \(b = a + k, \) and \( c = a + 2k.\) Thus,\[ d = a^2 + (a+k)^2 + (a+2k)^2 \].

Step 3: Expand and simplify the equation for d

Expand and simplify:\[ d = a^2 + (a+k)^2 + (a+2k)^2 \]\[ = a^2 + (a^2 + 2ak + k^2) + (a^2 + 4ak + 4k^2) \]\[ = 3a^2 + 6ak + 5k^2 \].

Step 4: Solve for d, knowing d = a.

Since \( d = a^2 + b^2 + c^2 \) and \( d = a + 3k \), equate:\[ a + 3k = 3a^2 + 6ak + 5k^2 \].

Step 5: Simplify and solve the equation

Rearranging gives:\[ 3a^2 + 6ak + 5k^2 - a - 3k = 0 \].Test plausible values for \( a \) and \( k \) that satisfy the distinction and equality. After testing, you find that for \( a = 0 \) and \( k = 1 \), the values satisfy the equation.

Step 6: Calculate a + b + c + d

For \( a = 0 \), \( k = 1 \), the sequence is \( 0, 1, 2, 3 \).Calculate:\( a + b + c + d = 0 + 1 + 2 + 3 = 6 \).But since we need to recheck for valid integers where \( d = a + 3k = 2 \), correct evaluation shows a different integer pairing.

Step 7: Double-check the values

Verify if other integer values satisfy the conditions better.For \( a = -1 \), \( k = 1 \), the A.P. is \( -1, 0, 1, 2 \) with:\[ (-1)^2 + 0^2 + 1^2 = 1 + 0 + 1 = 2,\]which matches \( a = 0 + a + 2k \). All values hold true.

Step 8: Verify and calculate a + b + c + d

For the valid sequence \( -1, 0, 1, 2 \), compute:\[ a + b + c + d = -1 + 0 + 1 + 2 = 2 \].

Key Concepts

Distinct Integers

In mathematics, integers are whole numbers that can be either positive, negative, or zero. They are distinct when each number is different from the others. In problems involving sequences, like Arithmetic Progressions (AP), it is crucial that the integers are distinct to ensure that each position in the sequence is occupied by a unique value.

For example, consider the sequence given in the exercise with numbers labeled as \(a, b, c, d\). Since these numbers are in an AP and must be distinct, no two values should be the same. This condition implies that the differences between each adjacent pair in the sequence are consistent and do not repeat any integer value.

Choosing distinct integers is important for maintaining the uniqueness of each number in the sequence, which is integral to solving equations and confirming accurate results when calculating cumulative sums.

Common Difference

In an arithmetic progression (AP), the common difference is the consistent value that separates consecutive terms in the sequence. It is denoted by \(k\) in mathematical notation.

Within the framework of the exercise, if the sequence given is \(a, a+k, a+2k, a+3k\), the number \(k\) represents how much each number increases from the previous one. This difference remains constant throughout the sequence.

Understanding the common difference is vital. It enables us to express any term in an AP in terms of the previous term. This uniformity is key when forming equations that utilize terms from arithmetic sequences. Recognizing this constant difference was particularly useful for solving equations in the exercise, and confirming that the derived sum matches the problem's requirements.

Summing Sequence

The act of summing sequences involves adding all the terms in a numeric sequence. In an arithmetic progression, the sum can often be represented by a formula, but simple addition suffice since sequences in problems are often short.

In the exercise, the sequence consists of four terms \(a, b, c, d\). When summed up to find their total, it provides useful information about the progression and helps verify if the conditions of the problem are satisfied.

In this case, the exercise required computing \(a + b + c + d\) for verification, ultimately arriving at the solution. This was achieved by confirming the distinct integers and ensuring that the arithmetic rule was upheld throughout the process. Understanding how to sum these sequences accurately is essential for resolving related problems effectively.

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the values of unknown variables that make an expression true.

In the current exercise, the equation to be solved was derived from setting the expression for \(d\) equal to the sum of squares of \(a, b, c\) and the arithmetic definition \(d = a + 3k\). This setup:

• \(d = a^2 + b^2 + c^2\)
• \(d = a + 3k\)

necessitated simplifying and rearranging to find valid integers that solve these simultaneous conditions.

Solving such equations often requires finding integer values for the sequence's starting term and the common difference. Testing possible values, as demonstrated, reveals the correct integers satisfying all equation parameters. Solving equations thus involves logical reasoning, trial, and error until a satisfactory solution confirms the arithmetic and problem statements.