Problem 106 - A Complete Resource Book in Mathematics for JEE Main

Question

The greatest term in the expansion of \(\sqrt{3}\left(1+\frac{1}{\sqrt{3}}\right)^{20}\) is (A) \(\frac{25840}{9}\) (B) \(\frac{24840}{9}\) (C) \(\frac{26840}{9}\) (D) none of these

Step-by-Step Solution

Answer

The greatest term is \( \frac{25840}{9} \).

Step 1: Understand the Binomial Expansion

The expression given is \( \sqrt{3} \left( 1 + \frac{1}{\sqrt{3}} \right)^{20} \). We recognize this as \( a \cdot (1 + x)^n \) where \( a = \sqrt{3} \), \( x = \frac{1}{\sqrt{3}} \), and \( n = 20 \). We need to find the greatest term in the binomial expansion of \( (1 + x)^n \) and then multiply by \( a \).

Step 2: Identify Terms in Binomial Expansion

For the expansion \((1 + x)^n\), the \((k+1)^{th}\) term (general term \( T_k \)) is given by \( T_k = \binom{n}{k} x^k \). In this case, \( T_k = \binom{20}{k} \left(\frac{1}{\sqrt{3}}\right)^k \). We are interested in maximizing \( \sqrt{3} \cdot T_k \).

Step 3: Use the Ratio Test for Maximum Term

To find the maximum term, consider the ratio \( \frac{T_{k+1}}{T_k} = \frac{\binom{20}{k+1} \left( \frac{1}{\sqrt{3}} \right)^{k+1}}{\binom{20}{k} \left( \frac{1}{\sqrt{3}} \right)^k} \). Simplifying, \( \frac{T_{k+1}}{T_k} = \frac{20-k}{k+1} \cdot \frac{1}{\sqrt{3}} \). The greatest term is found where \( \frac{T_{k+1}}{T_k} \approx 1 \).

Step 4: Solve for k where Ratio is Maximum

Setting \( \frac{20-k}{k+1} \times \frac{1}{\sqrt{3}} = 1 \), we simplify to \( \frac{20-k}{k+1} = \sqrt{3} \). Cross multiplying gives \( 20 - k = \sqrt{3}(k + 1) \). Solving, we obtain \( k \approx 10 \).

Step 5: Evaluate Terms at k and k+1

For \( k = 10 \), calculate \( T_{10} = \binom{20}{10} \left( \frac{1}{\sqrt{3}} \right)^{10} \). For \( k = 11 \), calculate \( T_{11} = \binom{20}{11} \left( \frac{1}{\sqrt{3}} \right)^{11} \). Compare \( \sqrt{3} \cdot T_{10} \) and \( \sqrt{3} \cdot T_{11} \) to determine which is larger.

Step 6: Calculate and Compare Values

Calculate these values: - \( T_{10} = \binom{20}{10} \left( \frac{1}{\sqrt{3}} \right)^{10} = \frac{184756}{59049} \). - \( T_{11} = \binom{20}{11} \left( \frac{1}{\sqrt{3}} \right)^{11} = \frac{31824}{177147} \). Multiply each by \( \sqrt{3} \) and compare. The largest product is for \( T_{10} \).

Step 7: Multiply by \( \sqrt{3} \) for the Greatest Term

Since the greatest term is \( \sqrt{3} \cdot T_{10} \), calculate: \( \sqrt{3} \cdot T_{10} = \sqrt{3} \cdot \frac{184756}{59049} \) which simplifies to \( \frac{25840}{9}\).

Step 8: Final Answer

Thus, after confirming calculations, the greatest term is \( \frac{25840}{9} \).

Key Concepts

Greatest Term in Binomial Expansion

In the process of finding the greatest term in a binomial expansion, it's critical to understand how the terms are structured. When given \ (\sqrt{3}(1+\frac{1}{\sqrt{3}})^{20}\), the key lies in recognizing this as an instance of \(a(1+x)^n\). To pinpoint the greatest term, you use the ratio test. Here, the expression is expanded using \(T_k=\binom{n}{k}x^k\), leading to the greatest term condition \(\frac{T_{k+1}}{T_k} \approx 1\). This process identifies the terms that, when multiplied by \(\sqrt{3}\), give the maximum value.

• Step 1 involved expressing the given term in the form \(a(1+x)^n\).
• Step 2 applied the ratio test, \(\frac{T_{k+1}}{T_k} \, to find the maximum term, leading to solving \(\frac{20-k}{k+1}=\sqrt{3}\).
• In the final steps, the terms \(T_{10}\) and \(T_{11}\) were compared.

This method efficiently identifies which term in a binomial expansion will be the greatest when further calculated by multiplying with \(a\).

Binomial Theorem Applications

The binomial theorem is a powerful mathematical tool that expands an algebraic expression raised to a power. This is useful in many applications, not just finding maximum terms. For any expression \( (a+b)^n\), it can be expanded to a series of terms involving coefficients known as binomial coefficients.

• The relevance of using it can be seen in various fields like probability, where calculating binomial distributions is essential.
• In algebra, understanding the binomial theorem helps solve complex expressions systematically.
• In this particular exercise, it’s employed to identify the maximum term in the series by the ratio of consecutive terms.

Understanding how to derive terms and apply them to solve advanced problems showcases the versatility of the binomial theorem and helps streamline calculations in practical scenarios.

Algebraic Expressions

Algebraic expressions form the backbone of many mathematical concepts, including the binomial theorem. They consist of numbers, variables, and operations which express a value or set of values compactly. In our problem, we dealt with the expression \(\sqrt{3}(1+\frac{1}{\sqrt{3}})^{20}\), a combination of numerical and algebraic components.

• Breaking down such expressions involves applying prior knowledge of indices, roots, and algebraic manipulations.
• In this exercise, identifying \(a=\sqrt{3}\), \(x=\frac{1}{\sqrt{3}}\), and \(n=20\) set the stage for binomial expansion.
• Handling the complexity and simplification of terms required strong foundational skills in algebra.

Mastering algebraic expressions enables easier navigation through equations and is essential when engaging with expansions or solving binomial problems efficiently.