Problem 98 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(S_{1}, S_{2}\) and \(S_{3}\) denote the sums up to \(n>1\) terms of three sequences in A.P. whose first terms are unity and common differences are in H.P. then \(n=\) (A) \(\frac{2 S_{3} S_{1}+S_{1} S_{2}+S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}\) (B) \(\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}+2 S_{2}+S_{3}}\) (C) \(\frac{2 S_{3} S_{1}-S_{1} S_{2}-S_{2} S_{3}}{S_{1}-2 S_{2}+S_{3}}\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is option (C).

Step 1: Understand the Problem

We are given three arithmetic progressions (A.P.) where each has the same first term, 1, and their common differences are in harmonic progression (H.P.). We denote the sum of their first n terms as \(S_1, S_2,\) and \(S_3\). We need to find the value of \(n\) in terms of these sums according to the given options.

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

The sum of the first \(n\) terms of an arithmetic progression with first term \(a\) and common difference \(d\) is given by:\[ S = \frac{n}{2} \times (2a + (n-1)d) \]Since the first term \(a = 1\), we have:\[ S = \frac{n}{2} \times (2 + (n-1)d) \] for each sequence.

Step 3: Relate Common Differences Using Harmonic Progression

In a harmonic progression, the reciprocals of the terms form an arithmetic progression. Therefore, if \(d_1, d_2, d_3\) are the common differences in H.P., then \(\frac{1}{d_1}, \frac{1}{d_2}, \frac{1}{d_3}\) form an A.P. Hence, \(\frac{2}{d_2} = \frac{1}{d_1} + \frac{1}{d_3}\).

Step 4: Setup Equations for Sums

Given the sequences, we have sums:\[ S_1 = \frac{n}{2}(2 + (n-1)d_1) \]\[ S_2 = \frac{n}{2}(2 + (n-1)d_2) \]\[ S_3 = \frac{n}{2}(2 + (n-1)d_3) \]

Step 5: Derive Expression Using Harmonic Relation

Use the relation from harmonic sequence: if \(d_1, d_2, d_3\) are in H.P., then \(n\) can be found by equating:\(2S_3S_1 = S_1S_2 + S_2S_3 + (S_1 - 2S_2 + S_3)n\).Rearrange to find \(n\):\(n = \frac{2S_3S_1 - S_1S_2 - S_2S_3}{S_1 - 2S_2 + S_3}\).

Step 6: Match Derived Expression with Options

Compare the derived expression for \(n\) with the listed options. The expression \(n = \frac{2S_3S_1 - S_1S_2 - S_2S_3}{S_1 - 2S_2 + S_3}\) matches option (C).

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is known as the 'common difference.' Understanding this concept is essential when working with sequences, as it helps to predict future terms and calculate the sum of the series.

For example, if you have a sequence starting at 2 with a common difference of 3, the sequence would be: 2, 5, 8, 11, and so on.
- **Key Characteristics of A.P.:**

• All terms are equidistant from each other, separated by the 'common difference.'
• The nth term of an A.P. can be expressed as: \(a_n = a + (n-1) \cdot d\), where \(a\) is the first term, and \(d\) is the common difference.
• Allows calculation of terms ahead in the sequence easily using its formula.

Understanding how different progressions work helps simplify many mathematical complexities and analyze relationships, much like in the original exercise where the common differences were set in a harmonic progression, providing a unique twist to the problem.

Sum of Sequence

Calculating the sum of a sequence in arithmetic progression is often required in mathematical problems and real-life applications. The sum of an arithmetic progression is found by taking the average of the first and last terms and multiplying by the number of terms.

The general formula for the sum of the first \(n\) terms in an arithmetic progression is:
\[S = \frac{n}{2} \times (2a + (n-1)d)\]
where:

• \(S\) is the sum of the sequence.
• \(n\) is the number of terms.
• \(a\) is the first term.
• \(d\) is the common difference.

This formula allows you to find the total of all numbers in an arithmetic series without having to add each one individually. For instance, in the given problem, knowing how to use this formula helps find \(S_1, S_2,\) and \(S_3\) for different sequences that have the same initial term and different common differences in H.P.
Understanding how to derive this sum is crucial in fields like finance to calculate series of payments or in physics when determining distance traveled in uniform acceleration scenarios.

Common Difference

The common difference in an arithmetic progression refers to the consistent interval that separates each term in the sequence. This constant is crucial because it influences the sequence's rate of increase or decrease.

The common difference \(d\) is calculated simply as the difference between any two successive terms:
\[ d = a_{n+1} - a_n \]
where:

• \(a_{n+1}\) is the subsequent term, and
• \(a_n\) is the current term.

In the original exercise problem, the common differences from three sequences were integrated into a harmonic progression, making them reciprocals of an arithmetic progression. This unique characteristic links arithmetic and harmonic progressions, showcasing their interconnectedness.

Understanding this concept deeply provides insight into how sequences progress and change, allowing for more accurate predictions and solutions in mathematical problems and practical applications.