Problem 25 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(n\) is positive integer and \(k\) is a positive integer not exceeding \(n\), then \(\sum_{k=1}^{n} k^{3}\left(\frac{C_{k}}{C_{k-1}}\right)^{2}\), where \(C_{k}={ }^{n} C_{k}\), is (A) \(\frac{n(n+1)(n+2)}{12}\) (B) \(\frac{n(n+1)^{2}(n+2)}{12}\) (C) \(\frac{n(n+1)^{2}(n+2)}{6}\) (D) none of these

Step-by-Step Solution

Answer

Option (B) \(\frac{n(n+1)^{2}(n+2)}{12}\).

Step 1: Understanding the Components

First, we need to understand that \(C_k\) represents the binomial coefficient \(\binom{n}{k}\), which is the number of ways to choose \(k\) elements from \(n\) elements. We are tasked with evaluating the expression \(\sum_{k=1}^{n} k^{3}\left(\frac{C_{k}}{C_{k-1}}\right)^{2}\).

Step 2: Simplifying the Coefficient Ratio

The ratio \(\frac{C_{k}}{C_{k-1}}\) can be simplified using the definition of binomial coefficients: \(\frac{\binom{n}{k}}{\binom{n}{k-1}} = \frac{n-k+1}{k}\).

Step 3: Applying the Ratio in the Sum

Insert the simplified ratio into the original expression: \(\sum_{k=1}^{n} k^{3}\left(\frac{n-k+1}{k}\right)^{2}\). This simplifies to \(\sum_{k=1}^{n} k^{3} \cdot \frac{(n-k+1)^2}{k^2}\), which can be further simplified to \(\sum_{k=1}^{n} k \cdot (n-k+1)^2\).

Step 4: Evaluating the Sum Expression

Expand \((n-k+1)^2\) to \((n-k+1)^2 = (n^2 - 2nk + n + k^2 - 2k + 1)\). Hence, the expression becomes \(\sum_{k=1}^{n} k(n^2 - 2nk + n + k^2 - 2k + 1)\).

Step 5: Separating and Solving the Summation

Split the summation into separate sums: \( n^2\sum_{k=1}^{n} k - 2n\sum_{k=1}^{n} k^2 + \sum_{k=1}^{n} k^3 + n\sum_{k=1}^{n} k - \sum_{k=1}^{n} 2k^2 + \sum_{k=1}^{n} k \).

Step 6: Solving Individual Sums

Use known formulas for the sums: \(\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\), \(\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\), and \(\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2\).

Step 7: Completing the Solution

Substitute the summation values back into the expression and simplify each term. The evaluation leads to cancelling terms, eventually simplifying to \(\frac{n(n+1)^{2}(n+2)}{12}\).

Step 8: Review Options

Compare the simplified expression \(\frac{n(n+1)^{2}(n+2)}{12}\) with the given options. It matches option (B), therefore this is the correct answer.

Key Concepts

Summation Techniques

Summation techniques are vital in mathematical problem-solving, particularly when dealing with series and sequences like the one given in the exercise. Understanding how to decompose and simplify sums is a powerful skill. In this problem, the summation \( \sum_{k=1}^{n} k^{3}\left(\frac{C_{k}}{C_{k-1}}\right)^{2} \) needs to be broken down into more manageable parts. The first step is recognizing the individual components of the sum and substituting them with simpler equivalents. For instance, expressing the binomial coefficient ratio \( \frac{C_{k}}{C_{k-1}} \) in terms of simple algebraic expressions makes the computation easier.

• Notice that simplifying \( \frac{C_{k}}{C_{k-1}} = \frac{n-k+1}{k} \) allows us to transform the original sum into terms like \( \sum k^{3}\). This makes further manipulation much simpler.
• Breaking up the expanded terms, as shown by \( n^2\sum_{k=1}^{n} k \), enables the evaluation using known summation formulas.

A strong grasp of different summation formulas, such as those for arithmetic and geometric series, is crucial for solving complex expressions efficiently. This step-by-step breakdown helps to simplify complicated calculations into a series of straightforward algebraic operations.

Algebraic Simplification

Algebraic simplification is a mathematical technique used to make complex expressions easier to understand and work with. In the given exercise, simplification is crucial for evaluating the expression and includes dealing with both algebraic terms and fractional expressions.To start, representing binomial coefficients using algebraic terms helps in simplifying ratios like \( \frac{C_{k}}{C_{k-1}} \). This fraction simplifies to \( \frac{n-k+1}{k} \), which is more straightforward to manipulate in further calculations.

• Once simplified, it's essential to apply algebraic identities to break complex expressions into more solvable components. For example, expanding \((n-k+1)^2\) into simple polynomials.
• From \( n^2 - 2nk + n + k^2 - 2k + 1 \), each term can then be isolated and handled separately, using known formulas and methods.

This approach reduces errors by using standard simplification techniques, ensuring that each part of the expression is calculated accurately and efficiently. Mastering these techniques enables you to tackle a broad range of algebraic problems with confidence.

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra, widely used to expand expressions raised to any power. For the exercise, understanding binomial coefficients - often involved with Pascal's Triangle - is essential. A binomial coefficient, denoted as \( C_k = \binom{n}{k} \), represents the number of ways to choose \( k \) items from \( n \), and plays a key role in expressions like \( \sum_{k=1}^{n} k^{3}\left(\frac{C_{k}}{C_{k-1}}\right)^{2} \).Applying the Binomial Theorem allows for the decomposition of polynomial expansions into simpler terms through binomial coefficients:

• Each coefficient reflects a term in the expansion of \((x + y)^n\).
• For example, simplifying expressions such as \( \binom{n}{k} \), using factorial methods, helps to integrate the coefficients into algebraic operations seamlessly.

This theorem is not only pivotal for expanding algebraic terms but also aids in deriving known summation formulas, as seen in the step-by-step solution of the given problem. Understanding these principles can greatly facilitate the evaluation of complex algebraic expressions in various mathematical contexts.