Let the coefficients of three consecutive terms in the binomial expansion of (1 + 2x)ⁿ be in the rat

Q: Wait, why is the coefficient of $T_{r+1}$ equal to ${}^nC_r 2^r$ and not just ${}^nC_r$?

This is the most common trap! In the expansion of $(1+ax)^n$, the general term is ${}^nC_r (1)^{n-r} (ax)^r$. The $a^r$ term is part of the coefficient. Since $a=2$ here, you must include $2^r$. Ignoring this will lead to an incorrect ratio.

Q: Is there a faster way to solve for $n$ and $r$ without using the combination formula?

You could expand the factorials, but it is prone to calculation errors. The property $\frac{{}^nC_r}{{}^nC_{r-1}} = \frac{n-r+1}{r}$ is a standard JEE tool. Mastering this identity saves time and reduces the complexity of your algebra significantly.

Q: Why do we assume the terms are $T_r, T_{r+1}, T_{r+2}$?

The problem states 'three consecutive terms'. We can label them however we like, but using $T_r, T_{r+1}, T_{r+2}$ aligns perfectly with the standard binomial coefficient notation ${}^nC_{r-1}, {}^nC_r, {}^nC_{r+1}$. It makes the algebra consistent and easy to track.

Question

Let the coefficients of three consecutive terms in the binomial expansion of $(1 + 2x)^n$ be in the ratio $2 : 5 : 8$. Then the coefficient of the term, which is in the middle of these three terms, is ______.

Sigma Percentile · Accepted Answer

### The Architecture of Binomial Expansion: A Journey into Coefficients

Welcome, future engineer. Today, we are not merely solving a problem about binomial coefficients; we are embarking on a journey to understand the hidden architecture of the Binomial Theorem.

When you look at an expression like $(1+2x)^n$, do not see it as a static string of characters. See it as a generator of patterns. The Binomial Theorem is one of the most elegant tools in our arsenal, but it demands precision.

One small slip—a forgotten power, a misplaced index—and the entire structure collapses. Let us walk through this problem with the care and rigor that the JEE Advanced demands.

### Phase 1: The Anatomy of the General Term

The first step in any binomial problem is to identify the 'General Term'. We know that for any expansion $(a+b)^n$, the general term $T_{r+1}$ is given by ${}^nC_r a^{n-r} b^r$.

In our specific case, the expansion is $(1+2x)^n$. Here, $a=1$ and $b=2x$. Therefore, the general term becomes:

$$T_{r+1} = {}^nC_r (1)^{n-r} (2x)^r$$

Now, pause here. This is where many students stumble. The coefficient of the term is not just ${}^nC_r$.

Because of the $2x$ term, we have a factor of $2^r$ that must be included. Thus, the coefficient $C_{r+1}$ is ${}^nC_r 2^r$.

If you miss this $2^r$, the ratio will be wrong, and the entire problem will lead you into a dead end. Always respect the variable's coefficient.

### Phase 2: The Logic of Consecutive Ratios

The problem gives us three consecutive terms with coefficients in the ratio $2:5:8$. Let us denote these coefficients as $C_r, C_{r+1}, C_{r+2}$.

Our goal is to translate this ratio into a system of equations. We have:

$$C_r = {}^nC_{r-1} 2^{r-1}, \quad C_{r+1} = {}^nC_r 2^r, \quad C_{r+2} = {}^nC_{r+1} 2^{r+1}$$

The ratio $C_r : C_{r+1} : C_{r+2} = 2 : 5 : 8$ implies two distinct relationships:

$$\frac{C_{r+1}}{C_r} = \frac{5}{2} \quad 	ext{and} \quad \frac{C_{r+2}}{C_{r+1}} = \frac{8}{5}$$

This is the heart of the problem. We are not just doing arithmetic; we are setting up a system of constraints that will force the values of $n$ and $r$ to reveal themselves.

### Phase 3: The Algebraic Dance

Let us tackle the first ratio:

$$\frac{C_{r+1}}{C_r} = \frac{{}^nC_r 2^r}{{}^nC_{r-1} 2^{r-1}} = \frac{5}{2}$$

Notice the beauty of the cancellation here. The $2^r$ and $2^{r-1}$ simplify to just $2$. We are left with:

$$2 	imes \frac{{}^nC_r}{{}^nC_{r-1}} = \frac{5}{2}$$

Now, we invoke the powerful identity $\frac{{}^nC_r}{{}^nC_{r-1}} = \frac{n-r+1}{r}$. Substituting this, we get:

$$2 	imes \frac{n-r+1}{r} = \frac{5}{2}$$

Cross-multiplying yields $4(n-r+1) = 5r$, which simplifies to:

$$4n - 9r = -4$$

This is our first anchor point. Now, we repeat the process for the second ratio:

$$\frac{C_{r+2}}{C_{r+1}} = \frac{{}^nC_{r+1} 2^{r+1}}{{}^nC_r 2^r} = \frac{8}{5}$$

Again, the powers of $2$ simplify to $2$. We use the identity $\frac{{}^nC_{r+1}}{{}^nC_r} = \frac{n-r}{r+1}$. Thus:

$$2 	imes \frac{n-r}{r+1} = \frac{8}{5}$$

Simplifying the $2$ and $8$ gives $\frac{n-r}{r+1} = \frac{4}{5}$. Cross-multiplying leads to $5(n-r) = 4(r+1)$, or:

$$5n - 9r = 4$$

### Phase 4: The Resolution

We now have a system of two linear equations:
(1) $4n - 9r = -4$
(2) $5n - 9r = 4$

Subtracting the first from the second is a stroke of elegance: $(5n - 9r) - (4n - 9r) = 4 - (-4)$, which gives us $n = 8$.

With $n=8$ in hand, we substitute back into the second equation:

$$5(8) - 9r = 4 \implies 40 - 9r = 4 \implies 9r = 36 \implies r = 4$$

We have found our variables! The middle term is $T_{r+1}$, which is $T_5$. Its coefficient is $C_{r+1} = {}^8C_4 	imes 2^4$.

Calculating ${}^8C_4$ is straightforward:

$$\frac{8 	imes 7 	imes 6 	imes 5}{4 	imes 3 	imes 2 	imes 1} = 70$$

And $2^4 = 16$. Finally, $70 	imes 16 = 1120$.

We have arrived at the destination. Remember, the math is not just about the final number; it is about the discipline of the process. You have successfully navigated the binomial expansion, handled the coefficients with care, and solved the system with precision.

The final answer is **1120**.

Animated Solution for Mathematics - Binomial Theorem: Let the coefficients of three consecutive terms in the binomial expansion of $(1 + 2 x)^{n}$ be in the ratio $2 : 5 : 8$ . Then the coefficient of the term, which is in the middle of these three terms, is ______.

Enter Numerical Value:

Visualized Solution

Identifying the General Term

Defining Three Consecutive Terms

Setting up the Ratios

First Ratio Equation

Applying Combination Formula (First Ratio)

Forming the First Equation

Second Ratio Equation

Applying Combination Formula (Second Ratio)

Forming the Second Equation

Solving for n and r

Calculating the Final Coefficient

Final Answer

The Sigma Insight: Properties of Binomial Coefficients

The Architecture of Binomial Expansion

Phase 1

Phase 2

Phase 3

Phase 4

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