Problem 5 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is: \(\quad\) [Sep. 03, 2020 (I)] (a) \(\frac{1}{6}\) (b) \(\frac{1}{5}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{7}\)

Step-by-Step Solution

Answer

The common difference of the A.P. is \( \frac{1}{6} \) (option (a)).

Step 1: Understanding the Problem

An A.P. (Arithmetic Progression) is given with the first term as 3. We need to find the common difference, given that the sum of the first 25 terms is equal to the sum of the next 15 terms.

Step 2: Using the Sum Formula for A.P.

The sum of the first \( n \) terms of an A.P. is given by the formula \( S_n = \frac{n}{2} (2a + (n-1)d) \), where \( a \) is the first term and \( d \) is the common difference. Here, \( a = 3 \).

Step 3: Calculate the First 25 Terms Sum

Using \( S_{25} = \frac{25}{2} (2 \cdot 3 + 24d) \), the sum of the first 25 terms is \( S_{25} = \frac{25}{2} (6 + 24d) = 25(3 + 12d) = 75 + 300d \).

Step 4: Calculate the Next 15 Terms Sum

The sum of terms from the 26th to the 40th term can be calculated by subtracting the sum of the first 25 terms from the sum of the first 40 terms: \( S_{15} = S_{40} - S_{25} \). Here \( S_{40} = \frac{40}{2} (2 \cdot 3 + 39d) = 20 (6 + 39d) = 120 + 780d \). So, \( S_{15} = (120 + 780d) - (75 + 300d) = 45 + 480d \).

Step 5: Setting Up the Equation

We know \( S_{25} = S_{15} \), thus \( 75 + 300d = 45 + 480d \).

Step 6: Solving the Equation

Rearrange to solve for \( d \): \( 75 - 45 = 480d - 300d \), or \( 30 = 180d \). Dividing both sides by 180 gives \( d = \frac{1}{6} \).

Step 7: Verification

Recalculate to confirm that \( 75 + 300 \times \frac{1}{6} = 45 + 480 \times \frac{1}{6} \); both sides equal 125 thus confirming the solution is correct.

Key Concepts

Common Difference

In an arithmetic progression (A.P.), the common difference is a crucial element that determines what each consecutive term in the sequence will be. Specifically, the common difference is the amount added to each term to get the next term in the sequence.
This concept is simple and follows a pattern:

• Imagine starting with a first term, denoted by \( a \).
• The second term will then be \( a + d \), where \( d \) is the common difference.
• For the third term, it would be \( a + 2d \), and it continues on similarly for further terms.

In the problem we are analyzing, although the first term is given as 3, determining the common difference was essential. By equating the sum of the first 25 terms of the progression to the sum of the next 15 terms, we could solve for \( d \). The solution reveals that the common difference for this sequence is \( \frac{1}{6} \).
The common difference essentially serves as the backbone of an A.P., dictating the rate of progression and overall structure of the sequence.

Sum of Terms

The sum of terms in an arithmetic progression is another pivotal concept when solving problems on A.P. It focuses on finding the sum of a specific number of terms in a sequence.
The formula for the sum of the first \( n \) terms of an arithmetic progression is given by:

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

Here, \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms to be summed.
In our exercise, we used this formula in two instances:

• First, to compute the sum of the first 25 terms, denoted as \( S_{25} \).
• Secondly, to find the sum of the first 40 terms, helping us derive the sum from the 26th to the 40th terms, denoted as \( S_{15} \).

Both calculations played a critical role in establishing an equation that was used to ultimately solve for the common difference. Remember that understanding which terms you need to sum is vital for applying the correct formula.

Arithmetic Sequence Formula

The arithmetic sequence formula is at the heart of understanding how arithmetic progressions work. This formula helps us figure out the exact value of any term in the sequence.
The general formula for finding the \( n^{th} \) term in an arithmetic progression is:

• \( a_n = a + (n-1) \times d \)

Where:

• \( a_n \) represents the \( n^{th} \) term.
• \( a \) is the first term.
• \( d \) is the common difference.
• \( n \) is the term number.

In the given problem, while the emphasis was on the sum of terms, understanding each term's contribution to these sums is based on this fundamental formula. For example:

• The 26th term would be calculated as \( a + 25d \).
• This is essential since both the sum formulas and equations derived rely on accurately identifying each term of the sequence based on its position number.

Thus, mastering the arithmetic sequence formula allows us to find any term in the sequence and all subsequent analyses build upon this basic knowledge.