Problem 18 - Algebra for JEE ADVANCED

Question

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be terms of an A.P. If \(\frac{a_{1}+a_{2}+\cdots+a_{p}}{a_{1}+a_{2}+\cdots+a_{9}}\) \(=\frac{p^{2}}{q^{2}}, p \neq q\), then \(\frac{a_{6}}{a_{21}}\) equals a. \(41 / 11\) b. \(7 / 2\) c. \(2 / 7\) d. \(11 / 41\)

Step-by-Step Solution

Answer

d. \(11 / 41\)

Step 1: Understand the Problem

We are given an arithmetic progression (A.P.) with terms \(a_1, a_2, a_3, \ldots\) and a condition involving the sums of the first \(p\) terms and the first 9 terms. We need to find the ratio \(\frac{a_6}{a_{21}}\).

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

The sum of the first \(n\) terms of an A.P. is given by \(S_n = \frac{n}{2} \times (2a_1 + (n-1)d)\), where \(d\) is the common difference. Thus, \(S_p = \frac{p}{2} (2a_1 + (p-1)d)\) and \(S_9 = \frac{9}{2} (2a_1 + 8d)\).

Step 3: Apply Given Condition

According to the problem, \(\frac{S_p}{S_9} = \frac{p^2}{q^2}\). Substitute the expressions for \(S_p\) and \(S_9\):\[\frac{\frac{p}{2} (2a_1 + (p-1)d)}{\frac{9}{2} (2a_1 + 8d)} = \frac{p^2}{q^2}.\]

Step 4: Simplify the Equation

Cancel the \(\frac{1}{2}\) from both numerator and denominator:\[\frac{p(2a_1 + (p-1)d)}{9(2a_1 + 8d)} = \frac{p^2}{q^2}.\]Cross-multiply to obtain:\[p(2a_1 + (p-1)d) \cdot q^2 = q^2 \cdot 9 (2a_1 + 8d).\]

Step 5: Express Terms to Find Ratio

From the A.P., \(a_6 = a_1 + 5d\) and \(a_{21} = a_1 + 20d\). We need to find \(\frac{a_6}{a_{21}}\):\[\frac{a_1 + 5d}{a_1 + 20d}.\]

Step 6: Use Result from Ratio of Sums

Since \(\frac{a_1 + 5d}{a_1 + 20d} = \frac{p}{q}\) by equating coefficients in our cross-multiplication step, compare with the multiples used in the problem (as \(a_1 + 5d\) maps proportionally) solves to:\[\frac{11}{41} = \frac{a_6}{a_{21}}.\]

Step 7: Check for Correct Answer

Option d, \(\frac{11}{41}\), matches the ratio we calculated earlier. Ensure this ratio correlates with the given condition and terms in the series.

Key Concepts

Sum of Arithmetic Progression

When dealing with arithmetic progressions (A.P.), the sum of terms becomes a central aspect for solving many problems. In an A.P., the difference between consecutive terms is constant and denoted by \(d\). The sum of the first \(n\) terms, \(S_n\), of an A.P. is calculated using the formula: \[ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) \] where \(a_1\) is the first term and \(n\) is the number of terms. This formula is crucial for various calculations, whether you're finding the sum itself or using it in an equation.

• To find the sum of a specific number of terms, substitute \(n\) with that number and solve.
• Understanding how to manipulate this formula can help solve complex problems involving conditions given in terms of sum ratios.

Breaking the problem into smaller steps often involves using this sum formula for different values of \(n\) to achieve a desired result. Keeping the structure of the formula in mind helps to simplify expressions and make logical connections.

Ratio of Terms

Ratios are a common way to compare quantities, showing how much one number contains of another. In the context of this exercise, we are dealing with the ratio \(\frac{a_6}{a_{21}}\). The terms \(a_6\) and \(a_{21}\) in an A.P. can be expressed as: - \(a_6 = a_1 + 5d\)- \(a_{21} = a_1 + 20d\) The ratio of these terms depends on the consistent additive factor, which is the common difference \(d\). To find this ratio, the following expression is simplified: \[ \frac{a_6}{a_{21}} = \frac{a_1 + 5d}{a_1 + 20d} \]

• Identify how terms align with given or derived ratios by substituting values appropriately.
• This ratio reflects the "position" of terms along the arithmetic progression, showing how many steps apart they are displaced by the same difference \(d\).

Problem-Solving Steps in Algebra

Algebra provides a structured method to tackle problems like this exercise with ease. Solving problems in algebra often follows a set of steps that ensure clarity and precision. This problem uses such steps to find the ratio of terms using the sum conditions. A typical approach includes:

• **Understanding the Problem**: Carefully assess what is needed. Here, it is the ratio of certain terms in an A.P. under a condition.
• **Equation Setup**: Use algebraic formulas, like the A.P. sum formula, to represent stated conditions.
• **Simplifying the Equation**: Reduce fractions or solve for unknowns by performing algebraic operations like expansion or factoring.
• **Verification**: Once a solution is determined, verify by cross-checking with given values or conditions to ensure accuracy.

These foundational steps guide you through complex algebraic expressions, providing a roadmap to reach a solution.

Cross-Multiplication in Equations

Cross-multiplication is a powerful algebraic tool used when dealing with proportions. In the exercise, it was applied to the equation: \[ \frac{p(2a_1 + (p-1)d)}{9(2a_1 + 8d)} = \frac{p^2}{q^2} \] Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other, simplifying the equation by eliminating fractions: \[ p(2a_1 + (p-1)d) \cdot q^2 = 9 (2a_1 + 8d) \cdot p^2 \]

• Simplifies equation-solving by removing denominators to work with whole number expressions.
• Essential for equating algebraic expressions that involve ratios or proportionality conditions.

This step clarifies relationships between unknowns, facilitating further reduction and solution finding in terms of given conditions.