The smallest natural number n, such that the coefficient of x in the expansion of (x² + 1/x³)^n is n

Q: Wait, why do we have two cases for $\binom{n}{r} = \binom{n}{23}$? Isn't $r$ just 23?

That is a classic trap! The binomial coefficient $\binom{n}{r}$ is symmetric. Think of Pascal's triangle—the values repeat. So, $\binom{n}{r} = \binom{n}{n-r}$. Always check both possibilities because the problem doesn't restrict $r$ to be 23; it only restricts the coefficient to be $\binom{n}{23}$.

Q: Why did we set $2n - 5r = 1$?

The problem asks for the coefficient of $x^1$. In our general term $T_{r+1}$, the power of $x$ is $2n - 5r$. To get $x^1$, that power must equal 1. It is the bridge between the algebra of the expansion and the specific term we need. We are essentially forcing the expansion to yield the term we want.

Q: Is it possible for $n$ to be smaller than 23?

Great question! If $n < 23$, then $\binom{n}{23}$ would technically be 0. However, the problem implies a non-zero coefficient, and we are looking for natural numbers. By solving the equations, we found $n=38$ and $n=58$, both of which are valid. Always let the algebra guide you, but keep the physical constraints of the binomial theorem in mind.

Question

The smallest natural number $n$, such that the coefficient of $x$ in the expansion of $(x^2 + \frac{1}{x^3})^n$ is $^nC_{23}$, is :

Options:

(A) 35

(B) 38

(C) 23

(D) 58

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The problem asks us to find the smallest natural number $n$ for the expansion of $(x^2 + \frac{1}{x^3})^n$ such that the coefficient of $x$ is $\binom{n}{23}$. This requires us to utilize the general term formula of the Binomial Theorem.

The general term $T_{r+1}$ is defined as:

$$T_{r+1} = \binom{n}{r} (x^2)^{n-r} (x^{-3})^r$$

By applying the laws of exponents, we simplify the expression for the power of $x$:

$$(x^2)^{n-r} \cdot (x^{-3})^r = x^{2n - 2r} \cdot x^{-3r} = x^{2n - 5r}$$

Thus, the general term simplifies to:

$$T_{r+1} = \binom{n}{r} x^{2n - 5r}$$

### The Governing Equations

To find the coefficient of $x^1$, we must set the exponent of $x$ equal to $1$:

$$2n - 5r = 1$$

Furthermore, the problem states that the coefficient of $x$ is $\binom{n}{23}$. From our general term, the coefficient is $\binom{n}{r}$, leading to the equality:

$$\binom{n}{r} = \binom{n}{23}$$

### Applying Binomial Symmetry

Recall the fundamental property of binomial coefficients: if $\binom{n}{a} = \binom{n}{b}$, then either $a = b$ or $a + b = n$. We must test both scenarios to find the possible values for $n$.

**Case 1: The Direct Path ($r = 23$)**

Substituting $r = 23$ into our exponent equation:

$$2n - 5(23) = 1$$
$$2n - 115 = 1$$
$$2n = 116 \implies n = 58$$

**Case 2: The Symmetric Path ($r = n - 23$)**

Substituting $r = n - 23$ into the exponent equation:

$$2n - 5(n - 23) = 1$$
$$2n - 5n + 115 = 1$$
$$-3n = -114 \implies n = 38$$

### Final Calculation

We have identified two potential values for $n$, which are $58$ and $38$. Since the question specifically asks for the smallest natural number $n$ that satisfies these conditions, we compare our results.

The smallest natural number is **$n = 38$**.

Animated Solution for Mathematics - Binomial Theorem: The smallest natural number $n$ , such that the coefficient of $x$ in the expansion of $(x^{2} + \frac{1}{x ^{3}})^{n}$ is $^{n} C_{23}$ , is :

Select Answer:

Visualized Solution

The Binomial Expression

The General Term Formula

Substitute into General Term

Simplify the Power of $x$

Set the Exponent to One

Relate to the Given Coefficient

Properties of Combinations

Case 1: $r = 23$

Case 2: $r = n - 23$

Final Conclusion

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

The Governing Equations

Applying Binomial Symmetry

Final Calculation

Similar Questions

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The smallest natural number n, such that the coefficient of x in the expansion of (x2+x31​)n is nC23​, is :

Select Answer:

Visualized Solution

The Binomial Expression

The General Term Formula

Substitute into General Term

Simplify the Power of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x

Set the Exponent to One

Relate to the Given Coefficient

Properties of Combinations

Case 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r=23

Case 2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } r=n−23

Final Conclusion

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

The Governing Equations

Applying Binomial Symmetry

Final Calculation

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