Problem 37 - A Complete Resource Book in Mathematics for JEE Main

Question

The first and last term of an A.P. are \(a\) and \(l\) respectively. If \(S\) is the sum of all the terms of the A.P. and the common difference is \(\frac{l^{2}-a^{2}}{k-(l+a)}\), then \(k\) is equal to (A) \(S\) (B) \(2 S\) (C) \(3 S\) (D) None of these

Step-by-Step Solution

Answer

k equals 2S, therefore, the answer is option (B).

Step 1: Understand the Given Terms

We are given an arithmetic progression (A.P.) where the first term is \(a\), the last term is \(l\), the sum of all terms is \(S\), and the common difference \(d\) is \(\frac{l^{2}-a^{2}}{k-(l+a)}\). We need to find the value of \(k\).

Step 2: Apply the Formula for the Sum of A.P.

The sum \(S\) of an A.P. with first term \(a\), last term \(l\), and \(n\) terms is given by the formula: \(S = \frac{n}{2} \cdot (a + l)\).

Step 3: Express Number of Terms in Terms of \(d\)

Since the common difference \(d\) is \(\frac{l^{2}-a^{2}}{k-(l+a)}\), we can express the number of terms \(n\) using the formula for the last term \(l = a + (n-1)\cdot d\). Then, solve it for \(n\): \(n = \frac{l-a}{d} + 1.\)

Step 4: Substitute \(d\) into \(n\)

Substitute \(d = \frac{l^{2}-a^{2}}{k-(l+a)}\) into the expression for \(n\):\[ n = \frac{l-a}{\frac{l^{2}-a^{2}}{k-(l+a)}} + 1 = \frac{(l-a)(k-(l+a))}{l^2-a^2} + 1 \]

Step 5: Simplify and Set Equation to Solve for \(k\)

The goal is to reconcile the expressions for \(S\). Substitute the expression for \(n\) into the sum formula \(S = \frac{n}{2} \cdot (a + l)\) and compare it with \(n\) found in Step 4. Simplify the expression:\[ S = \frac{1}{2} \left(\frac{(l-a)(k-(l+a))}{l^2-a^2} + 1\right) (a + l) \]

Step 6: Solve for \(k = 2S\)

Matching the expressions for \(S\) derived in Step 5 implies \(k = 2S\) must hold in order for the equations to be satisfied, making option (B) correct.

Key Concepts

Common Difference

In an arithmetic progression (A.P), each term after the first is obtained by adding a constant value known as the "common difference" to the previous term. Understanding this concept is central to grasping A.P. sequences:

• The common difference is denoted by the letter \( d \).
• If the first term of an A.P is \( a \) and the second term is \( a + d \), then the sequence continues as \( a, a + d, a + 2d, a + 3d, \ldots \).
• In any arithmetic sequence, every pair of consecutive terms have a difference of \( d \).

In the problem above, the common difference \( d \) is given by \( \frac{l^2 - a^2}{k - (l + a)} \). This expression shows the relationship between the first term \( a \), the last term \( l \), and an unspecified constant \( k \). To tackle such problems, we need to often solve for \( d \) using other provided information, as it is essential for uncovering underlying sequence qualities.

Sum of Arithmetic Series

The sum of an arithmetic series is the total of all terms from the first to the last term in the sequence. The formula for calculating this sum \( S \) is very helpful:

• For an A.P with \( n \) terms, first term \( a \), and last term \( l \), the sum is given by:\[ S = \frac{n}{2} \cdot (a + l) \]
• This formula effectively says to multiply the mean of the first and last terms by the number of terms in the series.
• The expression \( \frac{n}{2} \cdot (a + l) \) reflects how the sum emerges from averaging extremities and multiplying by the count of terms.

In our exercise, understanding how to derive and apply this formula is crucial for solving for unknown terms like \( k \). It's particularly useful when we need to reconcile different expressions linking \( S \). By leveraging the known values or expressions for terms, we can simplify and solve complex equations.

Number of Terms in a Series

Identifying the number of terms, or \( n \), within an arithmetic series is essential when calculating other aspects like the total sum. Here's how it can be determined:

• To find \( n \), the expression for the \( n \)-th term is used: \( l = a + (n - 1) \cdot d \).
• Rearranging this, it becomes \( n = \frac{l - a}{d} + 1 \), allowing us to calculate \( n \) if \( a \), \( l \), and \( d \) are known.
• This formula reflects that \( n \) depends directly on the difference between the last and first term, adjusted by the common difference.

In the given problem, this becomes particularly relevant as we need to equate various expressions involving \( n \) to solve for \( k \). Knowing how to determine \( n \) empowers us to substitute and manipulate equations effectively to extract desired outcomes.