If the sum of the coefficients of all the positive even powers of x in the binomial expansion of (2x

Q: Wait, why is the power of $x$ always even? $30-4r$ is even, but what if $r$ is not an integer?

Great observation! In the binomial expansion of $(a+b)^n$, the index $r$ must be an integer ranging from $0$ to $n$. Since $r$ is an integer, $4r$ is always even. Subtracting an even number ($4r$) from an even number ($30$) will always result in an even number. Thus, the power of $x$ is guaranteed to be even for all valid terms.

Q: Why can we just set $x=1$ to find the sum of all coefficients?

Think about the expansion $(a+b)^n = \sum \binom{n}{r} a^{n-r} b^r$. If we set the variable part of the terms to $1$, the expression becomes the sum of the coefficients $\binom{n}{r} \times (\text{constants})$. By setting $x=1$ in $(2x^3 + \frac{3}{x})^{10}$, we are essentially evaluating the sum of all coefficients $\binom{10}{r} 2^{10-r} 3^r$ because the $x$ terms become $1$.

Q: Why do we subtract the terms for $r=8, 9, 10$ instead of just calculating $r=0$ to $7$?

It is a matter of efficiency! Calculating eight terms ($r=0$ to $7$) is tedious and prone to arithmetic errors. Calculating only three terms ($r=8, 9, 10$) and subtracting them from the total sum is much faster. In competitive exams like JEE, choosing the path of least resistance is a vital skill.

Question

If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $\left( 2x^3 + \frac{3}{x} \right)^{10}$ is $5^{10} - \beta \cdot 3^9$, then $\beta$ is equal to ____.

Sigma Percentile · Accepted Answer

### Analyzing the Setup Imagine standing before the expression $(2x^3 + \frac{3}{x})^{10}$. It looks intimidating, a fortress of exponents and coefficients. But in the world of binomial expansion, every fortress has a key. Our goal is to find the sum of coefficients for all positive even powers of $x$. ### The Master Key To unlock any term in a binomial expansion, we rely on the general term formula: $$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$ Here, $n=10$, $a=2x^3$, and $b=\frac{3}{x}$. When we substitute these, we get: $$T_{r+1} = \binom{10}{r} (2x^3)^{10-r} \left(\frac{3}{x}\right)^r$$ By separating the constants from the variables, we find the power of $x$ is $30 - 4r$. This is the heartbeat of our problem. ### The Detective Work We have two constraints: the power must be even, and it must be positive. As we discussed, $30 - 4r$ is always even for any integer $r$. The real challenge is the positivity constraint: $30 - 4r > 0$. This simplifies to $4r < 30$, or $r < 7.5$. Since $r$ must be an integer, our valid range is $r \in \{0, 1, 2, 3, 4, 5, 6, 7\}$. ### The Elegant Shortcut Now, we could calculate each of these eight terms, but that is the long road. Instead, let us use the 'Total Sum' trick. If we set $x=1$ in our original expression $(2x^3 + \frac{3}{x})^{10}$, we get: $$(2(1)^3 + 3)^{10} = (2+3)^{10} = 5^{10}$$ This $5^{10}$ is the sum of **all** coefficients from $r=0$ to $r=10$. To find our required sum, we simply take this total and subtract the 'forbidden' terms—those where $r > 7.5$, which are $r=8, 9,$ and $10$. ### The Final Calculation For $r=8$, the coefficient is: $$\binom{10}{8} 2^2 3^8 = 45 \cdot 4 \cdot 3^8 = 180 \cdot 3^8 = 60 \cdot 3^9$$ For $r=9$, the coefficient is: $$\binom{10}{9} 2^1 3^9 = 10 \cdot 2 \cdot 3^9 = 20 \cdot 3^9$$ For $r=10$, the coefficient is: $$\binom{10}{10} 2^0 3^{10} = 1 \cdot 1 \cdot 3^{10} = 3 \cdot 3^9$$ Adding these gives $(60+20+3) \cdot 3^9 = 83 \cdot 3^9$. Thus, our sum is $5^{10} - 83 \cdot 3^9$. Comparing this to the given $5^{10} - \beta \cdot 3^9$, we find **$\beta = 83$**. We have conquered the problem with elegance and precision.

Animated Solution for Mathematics - Binomial Theorem: If the sum of the coefficients of all the positive even powers of $x$ in the binomial expansion of $(2 x^{3} + \frac{3}{x})^{10}$ is $5^{10} - β \cdot 3^{9}$ , then $β$ is equal to ____.

Enter Numerical Value:

Visualized Solution

The Binomial Expression

General Term Formula

Substituting the Values

Separating Constants and Variables

Simplifying the Power of $x$

Analyzing the Power of $x$

Constraint for Positive Power

Defining the Required Sum

Calculating Total Sum of Coefficients

Identifying Excluded Terms

Coefficient for $r = 8$

Coefficient for $r = 9$

Coefficient for $r = 10$

Summing the Excluded Terms

Final Comparison

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

The Master Key

The Detective Work

The Elegant Shortcut

The Final Calculation

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