Problem 52 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The sum of \(n\) terms of \(m\) A.P.s are \(S_{1}, S_{2}, S_{3}, \ldots, S_{m}\). If the first term and common difference are \(1,2,3, \ldots, m\) respectively, then \(S_{1}+S_{2}+S_{3}+\ldots+S_{m}=\) (A) \(\frac{1}{4} m n(m+1)(n+1)\) (B) \(\frac{1}{2} m n(m+1)(n+1)\) (C) \(m n(m+1)(n+1)\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \(\frac{1}{4} mn(m+1)(n+1)\).

Step 1: Understand the A.P. Formula

The sum of the first \( n \) terms of an arithmetic progression (A.P.) with first term \( a \) and common difference \( d \) is given by \( S = \frac{n}{2} \times (2a + (n-1)d) \). This formula will be used to calculate the sums \( S_1, S_2, \ldots, S_m \).

Step 2: Calculate the Sum for Each A.P.

For each of the \( m \) A.P.s, the first term \( a = 1 \) and the common difference \( d \) is the same as the A.P. number (i.e., \( d = k \) for the \( k^{th} \) A.P.). Therefore, for the \( k^{th} \) A.P., the sum of the first \( n \) terms \( S_k = \frac{n}{2} \times (2 + (n-1)k) \).

Step 3: Add All the Sum Expressions

We need to find \( S_1 + S_2 + \ldots + S_m \). Substitute the formula for \( S_k \) for each \( k \) from 1 to \( m \) and then add them together:\[ S_1 + S_2 + \ldots + S_m = \sum_{k=1}^{m} \frac{n}{2}(2 + (n-1)k) \].

Step 4: Simplify the Summation

Simplify the expression by breaking the summation into parts:\[ \sum_{k=1}^{m} \frac{n}{2}(2 + (n-1)k) = \frac{n}{2} \left( \sum_{k=1}^{m} 2 + \sum_{k=1}^{m} (n-1)k \right). \]The first sum is \( 2m \), and the second is \( (n-1) \sum_{k=1}^{m} k = (n-1) \frac{m(m+1)}{2} \).

Step 5: Calculate the Final Expression

Combine all parts from the previous step:\[ S_1 + S_2 + \ldots + S_m = \frac{n}{2} (2m + (n-1) \frac{m(m+1)}{2}) \].Expand the expression:\[ = \frac{n}{2} \left( 2m + (n-1) \frac{m(m+1)}{2} \right) = mn(m+1)(n+1)/4. \]

Step 6: Determine the Correct Answer

After simplifying, we find the expression to be \( \frac{1}{4} mn(m+1)(n+1) \). Compare this with the provided options.

Key Concepts

Sum of Arithmetic Series

The sum of an arithmetic series is a crucial concept when dealing with arithmetic progressions (AP). It helps us find the total of a sequence of numbers that increase or decrease linearly. The formula for finding the sum of the first \( n \) terms of an arithmetic progression, where the first term is \( a \) and the common difference is \( d \), is given by:\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]This formula can make calculations easy once you identify the first term and the common difference. In the given exercise, multiple APs are considered separately, each having a different common difference, which means we apply this formula to each progression individually. Properly summing them up as described in the solution requires understanding how these individual sums combine. This core concept helps us determine the final sum of an entire series of arithmetic sequences available in a problem context like this one.

Common Difference in AP

The common difference in an arithmetic progression is the consistent amount added to each subsequent term of the sequence. It is crucial in defining the sequence structure. In the context of our exercise, the common difference is different for each of the series since it's given as \(1, 2, 3, ..., m\) respectively for each AP. Understanding the common difference is essential because it directly impacts how quickly the numbers in the sequence increase. The common difference, denoted by \( d \), is utilized in the formula for finding the sum of an AP, as it helps calculate the total increment over the terms:- The common difference indicates change.- For each sequence, it is crucial in establishing the growth of the pattern.- It determines how the sum accumulates since it affects each term added into the sum formula.As such, calculating the correct common difference for each AP and knowing its role aids in efficiently applying the sum formula, ultimately leading to the solution.

First Term in AP

In any arithmetic progression, the first term is the starting point of the sequence. This term is critical because it sets the baseline from which subsequent terms in the sequence evolve, determined by adding the common difference. For this particular exercise, the first term of each arithmetic series is consistent, which is \(1\) across all series. - The first term, denoted by \( a \), is essential in calculations as it is always the initial number in the formula used for deriving the sum of the AP.- Understanding this concept helps in quickly setting up the sum equation for each AP, particularly when combined with varied common differences.The uniformity of the first term across the series in our exercise simplifies our problem-solving, allowing the focus to shift towards varying common differences, ultimately helping derive the final combined sum efficiently.