Question

Let \(V_{r}\) denote the sum of the first \(r\) terms of an arithmetic progression (A.P.) whose first term is \(r\) and the common difference is \((2 r-1) .\) Let $$T_{r}=V_{r+1}-V_{r}-2 \text { and } Q_{r}=T_{r+1}-T_{r} \text { for } r=1,2, \ldots$$ Which one of the following is a correct statement? (A) \(Q_{1}, Q_{2}, Q_{3}, \ldots\) are in A.P. with common difference 5 (B) \(Q_{1}, Q_{2}, Q_{3}, \ldots\) are in A.P. with common difference 6 (C) \(Q_{1}, Q_{2}, Q_{3}, \ldots\) are in A.P. with common difference 11 (D) \(Q_{1}=Q_{2}=Q_{3}=\ldots\)

Short answer

Option (D) is correct, as \(Q_{1}=Q_{2}=Q_{3}=\ldots=0\), indicating that all terms are equal and thus not forming a progression with a non-zero common difference.

Step-by-step solution

Determine the formula for the sum of the first r terms (Vr)

In an arithmetic progression with first term 'a' and common difference 'd', the sum of the first r terms is given by the formula: \(V_r = \frac{r}{2}[2a + (r - 1)d]\). Here, 'a' is r and 'd' is (2r - 1), so we substitute those values to get the formula for Vr specific to this sequence: \(V_r = \frac{r}{2}[2r + (r - 1)(2r - 1)]\).

Find Tr using Vr

We have the definition of Tr as \(T_r=V_{r+1}-V_{r}-2\). We will substitute the formula found for Vr into this equation to express Tr in terms of r.

Simplify the expression for Tr

By substituting the formula from Step 1 into the definition of Tr from Step 2 and simplifying, we obtain an expression for Tr solely in terms of r.

Calculate Qr

The definition of Qr is given by \(Q_{r}=T_{r+1}-T_{r}\). We need to substitute the simplified Tr formula from Step 3 to find the value of Qr.

Identify the pattern in Qr

By calculating the first few values of Qr using the formula from Step 4, a pattern can be observed to determine whether the sequence of \(Q_{1}, Q_{2}, Q_{3},\ldots\) is an arithmetic progression and, if so, what the common difference is.

Verify the correct option

Based on the pattern identified in Step 5, we compare the actual behavior of Qr with the options given to ascertain the correct statement.

Key concepts

Sum of Arithmetic Series

Understanding the sum of an arithmetic series is crucial for not only algebra but also for higher math, including JEE Advanced mathematics. In an arithmetic series, we deal with a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term.

The sum of the first 'n' terms of an arithmetic series can be found using the formula:\[\begin{equation}S_n = \frac{n}{2}(a_1 + a_n)\end{equation}\]where \(S_n\) is the sum of the first 'n' terms, \(a_1\) is the first term, and \(a_n\) is the \(n\)th term. An alternative formula is:\[\begin{equation}S_n = \frac{n}{2}(2a_1 + (n - 1)d)\end{equation}\]In this formula, 'd' represents the common difference. This method is particularly helpful when we do not know the last term but we do know the common difference and the number of terms.

For example, in the problem provided, the series has its first term as 'r' and the common difference as \((2r - 1)\). The sum of the first 'r' terms, therefore, uses the second formula and is tailored to include the specifics of this series.

Arithmetic Sequence Formula

The arithmetic sequence formula provides us with a direct approach to find any term in an arithmetic sequence. It is one of the key formulas students need to master for competitive exams like JEE Advanced mathematics. The formula for the \(n\)th term (\(a_n\)) of an arithmetic sequence is:\[\begin{equation}a_n = a_1 + (n - 1)d\end{equation}\]Here, \(a_1\) is the first term and 'd' is the common difference. The formula illustrates how every term is the sum of the first term and the product of the common difference and one less than the term’s position.

Applying this formula to the exercise in question, if we wish to find the rth term of the given arithmetic progression, we simply replace \(a_1\) with \(r\) and 'd' with \((2r - 1)\) to get the formula tailored to this specific sequence. It streamlines the process of identifying patterns and calculating subsequent terms without having to list the entire sequence, a technique that can save valuable time during an exam.

Arithmetic Series Pattern

Recognizing the arithmetic series pattern is critical for analyzing sequences and series. In an arithmetic series, since the difference between consecutive terms is constant, this leads to a distinctive linear pattern when plotting the terms on a graph. Patterns within arithmetic sequences and series can also manifest in the behavior of various operations on those terms, such as the differences between them, which is a focus of the given exercise.

To identify a pattern in the higher-order differences, we perform sequential subtractions, a technique often tested in JEE Advanced mathematics. When we calculate \(T_r=V_{r+1}-V_{r}-2\) for our example problem, we're looking for a regular pattern in these differences. If a constant value emerges through these subtractions, we can conclude that the higher-order differences form an arithmetic sequence on their own. This type of pattern recognition is key in mathematics as it allows us to extrapolate behaviors and predict future terms or outcomes with accuracy and efficiency.

JEE Advanced Mathematics

JEE Advanced Mathematics encompasses a wide variety of topics, including the in-depth study of arithmetic progressions and series. It requires students to have a strong grasp of concepts and formulas along with the ability to identify patterns and apply logical reasoning.

In the context of the question provided, it tests the understanding of arithmetic series by exploring not just the direct calculation of terms but also the relationship between successive terms and their differences. This leverages the students' analytical skills and their ability to manipulate sequences and series algebraically.

Questions like the one presented often appear in the JEE Advanced exam, and they assess a student’s proficiency in converting a real-world or theoretical problem into a mathematical model — in this case, an arithmetic progression — and then analyzing it to find a logical conclusion. Developing these skills is fundamental for success in the exam and for pursuing further studies in mathematics and related disciplines.