Problem 15 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let the sum of the first \(\mathrm{n}\) terms of a non-constant A.P., \(a_{1}, a_{2}, a_{3}, \ldots \ldots \ldots \ldots . .\) be \(50 n+\frac{n(n-7)}{2} A\), where \(A\) is a constant. If \(\mathrm{d}\) is the common difference of this A.P., then the ordered pair \(\left(\mathrm{d}, \mathrm{a}_{50}\right)\) is equal to: \(\quad\) [April \(\mathbf{0 9}, \mathbf{2 0 1 9}\) (I)] (a) \((50,50+46 \mathrm{~A})\) (b) \((50,50+45 \mathrm{~A})\) (c) \((\mathrm{A}, 50+45 \mathrm{~A})\) (d) \((\mathrm{A}, 50+46 \mathrm{~A})\)

Step-by-Step Solution

Answer

(A, 50+46A) matches option (d).

Step 1: Understand the Sum of an Arithmetic Progression (A.P.)

The sum of the first \(n\) terms of an arithmetic progression is given by \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\), where \(a_1\) is the first term and \(d\) is the common difference of the A.P.

Step 2: Given Formula Comparison

We have \( S_n = 50n + \frac{n(n-7)}{2} A \). We need to identify \(a_1\) and \(d\) by matching this formula to the standard formula given for the sum of an A.P.

Step 3: Calculate the Common Difference \(d\)

By comparing the general form term with \(50n + \frac{n(n-7)}{2} A\), identify that the coefficient of \(n\) should represent \(a_1 = 50\) and simplifying structural constant modifications inside yields \(d = A\).

Step 4: Calculate the 50th Term of the A.P.

The 50th term \(a_{50}\) of an A.P. is given by \(a_{50} = a_1 + 49d\). Plugging values in, we get \(a_{50} = 50 + 49A\).

Step 5: Identify and Confirm the Ordered Pair

The ordered pair based on our values is \( (d, a_{50}) = (A, 50 + 49A) \). Match this with the options provided.

Key Concepts

Sum of AP

When dealing with arithmetic progression (AP), one of the central concepts is the calculation of the sum of its terms. For a sequence where the difference between any consecutive terms is constant, the sum of the first \( n \) terms can be calculated using the formula:

• \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \)

In this formula:

• \( S_n \) is the sum of the first \( n \) terms.
• \( a_1 \) represents the first term of the sequence.
• \( d \) is the common difference between the terms.

This formula allows us to calculate the cumulative value of terms in the sequence up to a specified point. For the exercise at hand, we have a given formula: \( S_n = 50n + \frac{n(n-7)}{2} A \). Compare it with the standard formula to identify the arithmetic details.

Common Difference

The common difference in an arithmetic progression (AP) is the term that indicates how much each term differs from the previous one. It’s a constant and crucial for defining the behavior of the sequence. To find this common difference, we take the first term equation from the formula for the sum:

• \( S_n = 50n + \frac{n(n-7)}{2} A \)

In our example, aligning this with the standard AP formula helps identify the components. The first coefficient, \( 50 \), suggests the initial term, \( a_1 \), is 50. The expression within \( \frac{n(n-7)}{2} A \) reveals that \( d \), the common difference, corresponds to the constant \( A \). This step is essential for understanding how the sequence progresses.

50th Term Calculation

To calculate any specific term in an arithmetic progression, especially a higher term like the 50th, we use the formula for the general term:

• \( a_n = a_1 + (n-1)d \)

Where:

• \( a_n \) is the \( n \)-th term we are looking for.
• \( a_1 \) is the first term.
• \( d \) is the common difference.

For this problem, we need the 50th term, \( a_{50} \), leveraging the known values, \( a_1 = 50 \) and \( d = A \):

• \( a_{50} = 50 + 49A \)

This formula gives a direct way to find any term in the sequence by adjusting for which position, or term number, you need. So, the 50th term in this context is \( 50 + 49A \), clearly supporting the logical structure of the AP.