Problem 7 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the sum of first 11 terms of an A.P., \(a_{1}, a_{2}, a_{3}, \ldots\) is 0 \(\left(a_{1} \neq 0\right)\), then the sum of the A.P., \(a_{1}, a_{3}, a_{5}, \ldots, a_{23}\) is \(k a_{1}\), where \(k\) is equal to : [Sep. 02, 2020 (II)] (a) \(-\frac{121}{10}\) (b) \(\frac{121}{10}\) (c) \(\frac{72}{5}\) (d) \(-\frac{72}{5}\)

Step-by-Step Solution

Answer

The value of \(k\) is \(-\frac{72}{5}\); option (d).

Step 1: Identify the Terms of the AP

The given arithmetic progression (A.P.) is defined as \(a_1, a_2, a_3, \ldots\) with a common difference \(d\). The formula for the \(n\)th term is given as \(a_n = a_1 + (n-1) \cdot d\).

Step 2: Use the Sum Formula for the First 11 Terms

The sum of the first 11 terms of an A.P. is given by the formula \(S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d)\). For the first 11 terms, \(n = 11\), and it's given that this sum is 0. Thus, we have:\[\frac{11}{2} \cdot (2a_1 + 10d) = 0\]Simplifying gives: \[2a_1 + 10d = 0\]\[a_1 + 5d = 0\] Solving for \(a_1\), we find \(d = -\frac{a_1}{5}\).

Step 3: Formulate the Given Logarithmic Progression

We need to find the sum of the sequence \(a_1, a_3, a_5, \ldots, a_{23}\). This sequence is every second term of the original A.P., forming a new A.P. with first term \(a_1\) and common difference \(2d\). Calculate the number of terms in this sequence starting from \(a_1\) to \(a_{23}\) by substituting into the sequence of odd-numbered terms: \(a_{2k-1} = a_1 + (2k-2)d\). For \(a_{23}=a_1+(23-1)d\).

Step 4: Derive the Common Difference for the New AP

For the new sequence \(a_1, a_3, a_5, \ldots, a_{23}\), the term \(a_{23}\) corresponds to the 12th term:\(2n - 1 = 23\) gives \(n = 12\).The number of terms in this sequence is therefore 12.

Step 5: Calculate the Sum of New AP

Use the sum formula for the first \(n\) terms for this new A.P.:\[S_n = \frac{n}{2} \cdot [2a_1 + (n-1) \cdot 2d] = \frac{12}{2} \cdot [2a_1 + 22d]\]\[S_{12} = 6 \times [2a_1 - \frac{22a_1}{5}]\]This simplifies to: \[S_{12} = 6 \times [2a_1 - \frac{22a_1}{5}] = 6 \times \frac{10a_1 - 22a_1}{5} = 6 \times \frac{-12a_1}{5}\]\[S_{12} = -\frac{72a_1}{5}\]Hence, comparing with the form \(ka_1\), \(k = -\frac{72}{5}\).

Step 6: Conclude the Solution

From the calculation, the value of \(k\) is \(-\frac{72}{5}\). Thus, the correct option is (d) \(-\frac{72}{5}\).

Key Concepts

Sum of an AP

The sum of an arithmetic progression (AP) is an important concept that allows you to find the total of all the terms in the sequence. An arithmetic progression is a sequence of numbers in which the difference of any two consecutive terms is always the same. This difference is called the common difference.

To calculate the sum of the first \(n\) terms of an AP, we use the formula:

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

Here, \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term of the sequence, \(n\) is the number of terms you're adding, and \(d\) is the common difference.

This formula gives you the ability to quickly find the total sum without adding each term individually. It's especially handy when dealing with long sequences because it reduces complex calculations to a simple multiplication and division method.
For example, as shown in the step-by-step solution, knowing that the sum of the first 11 terms is zero and using this formula can help you calculate the common difference \(d\) in the original sequence.

Nth term of an AP

Understanding the concept of the "Nth term" of an AP is crucial for identifying any term in the sequence. The general formula for finding the \(n\)th term of an arithmetic progression is:

• \( a_n = a_1 + (n-1) \cdot d \)

This formula tells you that the \(n\)th term (\(a_n\)) is calculated by adding the first term (\(a_1\)) and the product of \((n-1)\) times the common difference \(d\).

Using this formula, you can find any term in the sequence by adjusting the value of \(n\). It enables you to quickly identify specific terms without listing each one by hand.
For example, to find the 23rd term in a given AP, plug \(23\) into the formula as \(n\). It helps you calculate precisely which term you seek in the sequence.

In the given problem, once the common difference \(d\) was identified, this formula helped determine how terms like \(a_{23}\) related to the new AP derived from odd terms of the original sequence.

Common difference

The common difference is a key feature in an arithmetic progression that defines the property and pattern of the sequence. Denoted as \(d\), it represents the constant amount by which each term in the sequence increases or decreases from the previous term.

If you know the first term and the common difference, you can generate the entire sequence. It essentially reflects the pattern governing the progression.

• Positive \(d\) means each term is larger than its predecessor.
• Negative \(d\) means each term is smaller.
• Zero \(d\) means all terms are the same.

In the solved exercise, knowing the sum of the first 11 terms was zero allowed us to derive \(d = -\frac{a_1}{5}\) which importantly maintains that consistent difference between terms and directly impacts how subsequent terms in the sequence are calculated.
This not only aids in finding the regular terms but also with subsequences like \(a_1, a_3, a_5, \ldots\) shared in the problem, the updated common difference \(2d\) guides the pattern of the newly formed sequence.