If x is positive, the first negative term in the expansion of (1 + x)²⁷/⁵ is (a) 6th term (b) 7th te

Q: Wait, why is the expansion infinite? Shouldn't it stop at some point?

That is a fantastic question! The binomial expansion $(1+x)^n$ only terminates if $n$ is a non-negative integer. Here, $n = \frac{27}{5}$, which is a rational number. When the exponent is not a positive integer, the series expansion continues indefinitely, creating an infinite sequence of terms.

Q: Why does the sign only depend on the numerator? What about $x^r$?

Great observation. The problem explicitly states that $x > 0$. Because $x$ is positive, any power of $x$ (i.e., $x^r$) will also be positive for any $r$. Similarly, $r!$ is always positive. Therefore, the only part of the general term that can change sign is the product of the factors in the numerator.

Q: If we found $r=7$, why is the answer the 8th term?

Remember the general term formula: $T_{r+1}$. The index $r$ starts from $0$ for the first term ($T_1$). So, when $r=1$, we are looking at the 2nd term ($T_2$). Following this logic, when $r=7$, we are looking at $T_{7+1}$, which is the 8th term. Always keep that $+1$ shift in mind!

Question

If $x$ is positive, the first negative term in the expansion of $(1 + x)^{27/5}$ is

Options:

(A) 6th term

(B) 7th term

(C) 5th term

(D) 8th term

Sigma Percentile · Accepted Answer

### Analyzing the Setup When you encounter an expression like $(1+x)^{\frac{27}{5}}$, your first instinct might be to look for a stopping point. However, because the exponent $n = \frac{27}{5}$ is not a positive integer, this expansion does not terminate. It stretches out into an infinite series. Our mission is to find the exact moment this series dips into negative territory. ### The Master Key: The General Term To navigate this infinite series, we rely on the general term formula: $$T_{r+1} = \frac{n(n-1)(n-2)\dots(n-r+1)}{r!} x^r$$ This formula is your best friend. It allows us to isolate any term in the sequence by simply choosing the value of $r$. Think of $r$ as a counter that tracks our position in the series. As we move from term to term, $r$ increases, and the product in the numerator evolves. ### The Detective Work: Analyzing the Sign We are given that $x > 0$. This is a crucial piece of information because it ensures that $x^r$ will always be positive. Since the denominator, $r!$, is also always positive, the sign of the entire term $T_{r+1}$ depends solely on the numerator product: $$P = n(n-1)(n-2)\dots(n-r+1)$$ We start with $n = 5.4$, which is positive. As we increase $r$, we multiply by factors that decrease by exactly $1$ at each step: $5.4, 4.4, 3.4, 2.4, 1.4, 0.4$. ### The Descent into the Negative Imagine standing on a number line. We start at $5.4$ and take a step of size $1$ for every increase in $r$. At $r=6$, the factor is $0.4$. We are still positive, but we are standing on the edge of a cliff. The very next step, at $r=7$, will take us to $0.4 - 1 = -0.6$. Mathematically, we want the last factor, $(n-r+1)$, to be less than zero. Substituting $n = 5.4$: $$5.4 - r + 1 < 0$$ $$6.4 - r < 0 \Rightarrow r > 6.4$$ Since $r$ must be an integer, the smallest value that satisfies this condition is $r=7$. ### Final Calculation We have found that $r=7$ is the first value that makes our product negative. However, the question asks for the term number. Our general term is defined as $T_{r+1}$. Substituting $r=7$, we get: $$T_{7+1} = T_8$$ Thus, the **8th term** is the first negative term in this infinite expansion. You have successfully navigated the infinite and solved the inequality!

Animated Solution for Mathematics - Binomial Theorem: If $x$ is positive, the first negative term in the expansion of $(1 + x)^{27/5}$ is

Select Answer:

Visualized Solution

Binomial Expansion for Rational Index

The General Term $T_{r + 1}$

What Determines the Sign?

The Product of Factors

Substituting the Value of $n$

Tracking Factors as $r$ Increases

Nearing the Negative Zone

Crossing the Zero Boundary

Setting up the Inequality

Substituting and Simplifying

Finding the Range for $r$

The Smallest Integer $r$

Which Term is It?

The Sigma Insight: Binomial Theorem for any Index

Analyzing the Setup

The Master Key

The Detective Work

The Descent into the Negative

Final Calculation

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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; } If x is positive, the first negative term in the expansion of (1+x)27/5 is