Problem 65 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If the ratio of the sum of \(m\) term and \(n\) terms of an A.P. be \(m^{2}: n^{2}\), prove that the ratio of its \(m\) th and \(n\) th terms will be \(2 m-1: 2 n-1\).

Step-by-Step Solution

Answer

The ratio of the sum of m terms and n terms of an A.P. is given as $m^{2}: n^{2}$. To prove the ratio of its m-th and n-th terms is $(2m - 1) : (2n - 1)$, we follow these steps: 1. Write the formula for the sum of m terms (S_m) and n terms (S_n) in an A.P. 2. Determine the given ratio of the sums: $S_{m} / S_{n} = m^{2} / n^{2}$. 3. Simplify the equation obtained in step 2. 4. Find the expressions for the m-th (T_m) and n-th terms (T_n) in the A.P. 5. Derive the required ratio between m-th and n-th terms: $(T_{m} / T_{n}) = (2m - 1) / (2n - 1)$. Following these steps, we prove that the ratio of the m-th and n-th terms is $(2m - 1) : (2n - 1)$.

Step 1: Formula for the sum of m terms and n terms in an A.P.

Let's denote the first term of the A.P. as 'a' and the common difference as 'd'. The formula for the sum of the first k terms in an A.P is given by: S_k = k/2 * [2a + (k - 1)d] Using this formula, we can write the sum of m terms (S_m) and n terms (S_n) as: S_m = m/2 * [2a + (m - 1)d] S_n = n/2 * [2a + (n - 1)d]

Step 2: Determine the given ratio of the sums

We are given that the ratio of the sum of m terms and n terms is m^2 : n^2. Let's write it down: S_m / S_n = m^2 / n^2 We'll substitute the expressions of S_m and S_n from Step 1: (m/2 * [2a + (m - 1)d]) / (n/2 * [2a + (n - 1)d]) = m^2 / n^2

Step 3: Simplify the equation

Now let's simplify the above equation by cancelling out the common terms: (2a + (m - 1)d) / (2a + (n - 1)d) = m^2 / n^2

Step 4: Find the expressions for m-th and n-th terms

To find the m-th and n-th terms, we'll use the expression for the k-th term in an A.P., which is given by: T_k = a + (k - 1)d Using this expression, we can write the m-th term (T_m) and n-th term (T_n) as: T_m = a + (m - 1)d T_n = a + (n - 1)d

Step 5: Derive the required ratio between m-th and n-th terms

We need to prove that the ratio of m-th and n-th terms is (2m - 1) : (2n - 1). So, we'll write the ratio of T_m and T_n: T_m / T_n = (a + (m - 1)d) / (a + (n - 1)d) Now, from the simplified equation in Step 3, we know that: (2a + (m - 1)d) / (2a + (n - 1)d) = m^2 / n^2 Comparing this equation with the ratio of T_m and T_n, we see that: T_m / T_n = (m^2 / n^2) * (2 / 2) = m^2 / n^2 Taking a square root on both sides, we get: sqrt(T_m / T_n) = sqrt(m^2 / n^2) Which simplifies to: (2m - 1) / (2n - 1) = T_m / T_n Thus, we have proven that the ratio of its m-th and n-th terms is (2m - 1) : (2n - 1).

Key Concepts

Sum of terms in an A.P.

Arithmetic Progression (A.P.) is one of the most fundamental concepts in mathematics, especially in sequences. A.P. is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. Calculating the sum of terms in an A.P. involves a specific formula: the sum of the first \(k\) terms, denoted as \(S_k\), can be calculated using the formula: \[S_k = \frac{k}{2} \times \left(2a + (k - 1)d\right)\]Here:

• \(a\) is the first term of the A.P.
• \(d\) is the common difference.
• \(k\) represents the number of terms whose sum needs to be calculated.

This formula helps us find the total value accumulated over a specific count of terms in the sequence. By substituting different values of \(k\), we can find sums for different segments of the progression, such as \(S_m\) for \(m\) terms and \(S_n\) for \(n\) terms. This concept is especially useful in solving problems where the comparison of sums over different sets of terms is required, like in the given exercise where ratios of sums are explored.

n-th term of an A.P.

In any arithmetic sequence, identifying a specific term is a common task. The formula for the n-th term of an A.P., denoted as \(T_n\), is crucial for pinpointing the exact position of a desired term. The formula for finding \(T_n\) is given by: \[T_n = a + (n - 1)d\]Here:

• \(a\) refers to the first term of the sequence.
• \(d\) is the common difference between terms.
• \(n\) is the term number you are trying to find.

This formula is not only essential for directly locating a term within the sequence but also aids in expressing complex relationships between different terms, as seen in problems that involve proving specific ratios. In our exercise, the n-th term formula helps in finding expressions for \(T_m\) and \(T_n\), the terms at positions \(m\) and \(n\) respectively, thus enabling further manipulation and proofs involving these terms.

Common difference in A.P.

The common difference in an A.P. is the cornerstone element that defines the uniformity of spacing between consecutive terms. Denoted by \(d\), this value remains constant throughout the sequence. The formula to determine the common difference is straightforward: \[d = T_2 - T_1\]This formula is derived from the general definition of an A.P., where each term increases by the same amount over the previous one. This consistency not only determines the overall trend of the sequence (whether it's increasing or decreasing) but is also essential in any formula related to an A.P. like the sum of terms and the n-th term. Understanding the common difference is vital for analyzing how sequences behave over extended terms and for solving complex problems, as it provides the baseline for any calculations involving positions or sums within the progression, such as in the exercise involving finding the ratio of specific terms.