Problem 66 - 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

The value of \(k\) is \(2S\) (Option B).

Step 1: Understanding the Arithmetic Progression (A.P.)

In an A.P., the first term is given by \(a\) and the last term by \(l\). The number of terms is represented by \(n\). The common difference is the difference between consecutive terms in the sequence.

Step 2: Sum of the A.P. Formula

The sum \(S\) of the first \(n\) terms of an A.P. is given by the formula \(S = \frac{n}{2} (a + l)\).

Step 3: Find the Common Difference

The problem states that the common difference \(d\) is given as \(d = \frac{l^2 - a^2}{k - (l + a)}\).

Step 4: Calculate Number of Terms in the A.P.

As the last term \(l\) is expressed as \(l = a + (n-1)d\), rearrange this to find \(n\) as \(n = \frac{l-a}{d} + 1\).

Step 5: Substitute Values of \(n\) and \(d\)

Substitute \(d\) from Step 3 into the expression for \(n\) in Step 4: \[ n = \frac{l-a}{\frac{l^2 - a^2}{k - (l + a)}} + 1 \]. Simplify this to: \[ n = \frac{(l-a)(k - (l + a))}{l^2-a^2} + 1 \].

Step 6: Simplify for \(k\) Using the Sum Formula

Given \(S = \frac{n}{2}(a + l)\), substitute \(n\) into this formula with the expression from Step 5. Simplify to solve for \(k\). This will involve simplifying the expression: \[ S = \frac{1}{2}(\frac{(l-a)(k - (l + a))}{l^2-a^2} + 1)(a + l) \]. Further simplification shows that \(k = 2S\).

Step 7: Verify Final Result for \(k\)

Upon completing algebraic simplifications, confirm that the result reduces cleanly to \(k = 2S\). This matches with option B.

Key Concepts

Sum of arithmetic progression

The sum of an arithmetic progression (A.P.) is a fundamental formula in mathematics. This formula allows us to calculate the total value when adding up all the terms in a given A.P. It's especially useful when dealing with large sequences. For an A.P. with the first term as \(a\), last term as \(l\), and \(n\) being the number of terms, the sum \(S\) is calculated using:
\[ S = \frac{n}{2} (a + l) \]This formula essentially takes the average of the first and last term and multiplies it by the number of terms to find the total sum. It's derived from the idea that each pair of terms equidistant from the start and end of the sequence adds up to the same value.

• Helps in calculating the total quickly.
• Always involves the first and last term for simplicity.

Understanding this formula allows you to approach any arithmetic progression problem with confidence. It is essential for deriving other related results like finding the number of terms or solving for unknowns within the sequence.

Common difference in A.P.

In an arithmetic progression (A.P.), the common difference \(d\) is the consistent interval between consecutive terms. This common difference can either be positive, negative, or zero, influencing how the sequence progresses. Knowing the common difference is crucial because it determines both the shape and direction of the sequence.
For instance, if the first term is \(a\) and the common difference is \(d\), the sequence looks like: \(a, a+d, a+2d, \ldots\). It can be calculated when the first and last terms, along with any term index, are known, or from the problem itself as given:
\[ d = \frac{l^2 - a^2}{k - (l + a)} \]This more complex expression for the common difference often requires simplification using algebraic techniques, aligning with other parts of the progression problem.

• Determines how the sequence increases or decreases.
• Can be derived from different aspects of the sequence.

Grasping the common difference's role helps clarify the full pattern of the progression, making it easier to solve for various elements like the number of terms or the sum.

Sequence of terms

A sequence of terms in an arithmetic progression (A.P.) follows a simple pattern defined by its first term and the common difference. Each term in an A.P. is derived from the formula:
\[ a_n = a + (n-1)d \]where \(a_n\) is the \(n^{th}\) term, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term in the sequence.
The terms of the sequence form a straight line graphically, making this type of progression linear. By applying the formula, you can predict or verify any term in the sequence.

• Each term is equidistant from its neighbors.
• Useful in finding any term directly if the first term and common difference are known.

The understanding of the sequence helps not just in direct calculations but also facilitates the discovery of hidden patterns or solving for unknowns within the broader algebraic context of progression problems. Recognizing how terms interrelate ensures comprehensive command over arithmetic progressions.