If for some positive integer n, the coefficients of three consecutive terms in the binomial expansio

Q: Wait, why do we use the ratio formula instead of just expanding the factorials like $$n! / (r!(n-r)!)$$?

Great question! While you *could* expand the factorials, it leads to a massive algebraic nightmare that is prone to calculation errors. The ratio formula $$\frac{^N C_r}{^N C_{r+1}} = \frac{r+1}{N-r}$$ is a 'shortcut' derived from the definition of combinations. In JEE Advanced, time is your currency—always look for the property that simplifies the expression before you start grinding through the algebra.

Q: How do I know if there is one largest coefficient or two?

It all depends on the parity of the power $$N$$. If $$N$$ is even, there is a single middle term at $$r = N/2$$. If $$N$$ is odd, the 'middle' is split between two terms at $$r = (N-1)/2$$ and $$r = (N+1)/2$$. Since our $$N$$ turned out to be 11 (odd), we have two equal largest coefficients. Always check the parity of $$N$$ before concluding!

Q: Why are the largest coefficients always in the middle?

Think of Pascal's Triangle. The values start at 1, increase steadily as you move toward the center, and then decrease symmetrically. This is because the binomial coefficient $$^N C_r$$ increases as $$r$$ approaches $$N/2$$. The 'peak' of this distribution is always at the center, making the middle terms the largest.

Question

If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5 : 10 : 14$, then the largest coefficient in this expansion is :

Options:

(A) 252

(B) 462

(C) 792

(D) 330

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are exploring the elegant architecture of the Binomial Theorem. We are given the expansion of $(1+x)^{n+5}$, where three consecutive coefficients exist in the ratio $5 : 10 : 14$.

Our mission is to uncover the value of $n$ and identify the largest coefficient in this expansion. Let $N = n+5$ represent the total power of the binomial.

### The Ratio Strategy

Let the three consecutive coefficients be $^{N}C_{r}$, $^{N}C_{r+1}$, and $^{N}C_{r+2}$. The ratio of the first two is $5:10$, which simplifies to $1:2$.

Using the property $\frac{^{N}C_{r}}{^{N}C_{r+1}} = \frac{r+1}{N-r}$, we substitute our values:

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

Cross-multiplying gives $2(r+1) = n+5-r$, which simplifies to the first linear equation:

$$3r = n+3$$

### The Algebraic Dance

Next, we use the ratio of the second and third coefficients, which is $10:14$, or $5:7$. Applying the ratio property $\frac{^{N}C_{r+1}}{^{N}C_{r+2}} = \frac{r+2}{N-(r+1)}$, we get:

$$\frac{r+2}{n+4-r} = \frac{5}{7}$$

Cross-multiplying yields $7(r+2) = 5(n+4-r)$. Expanding this results in $7r + 14 = 5n + 20 - 5r$, which rearranges to:

$$12r = 5n + 6$$

We now have a system of two equations: $3r = n+3$ and $12r = 5n + 6$. Multiplying the first equation by $4$ gives $12r = 4n + 12$.

Substituting this into the second equation, we find $4n + 12 = 5n + 6$, which leads us directly to **$n = 6$**.

### The Symmetry of Pascal's Triangle

With $n=6$, our expansion becomes $(1+x)^{11}$. In the expansion of $(1+x)^{N}$, the coefficients increase until they reach the middle and then decrease.

Because $N=11$ is an odd number, there are two middle terms. These occur at $r = \frac{N-1}{2}$ and $r = \frac{N+1}{2}$, which correspond to $r=5$ and $r=6$.

The largest coefficients are $^{11}C_{5}$ and $^{11}C_{6}$. Due to the symmetry of binomial coefficients, these two values are identical.

We calculate $^{11}C_{6}$ as follows:

$$^{11}C_{6} = \frac{11!}{6!5!} = \frac{11 	imes 10 	imes 9 	imes 8 	imes 7}{5 	imes 4 	imes 3 	imes 2 	imes 1}$$

After cancellation, we arrive at $11 	imes 3 	imes 2 	imes 7 = 462$. The largest coefficient in the expansion is **462**.

Animated Solution for Mathematics - Binomial Theorem: If for some positive integer $n$ , the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n + 5}$ are in the ratio $5 : 10 : 14$ , then the largest coefficient in this expansion is :

Select Answer:

Visualized Solution

Problem Statement

Consecutive Coefficients

Ratio Formula

First Linear Equation

Second Ratio Setup

Second Linear Equation

Solving for $n$

Total Number of Terms

Largest Coefficient Logic

Calculating $^{11} C_{6}$

The Sigma Insight: Properties of Binomial Coefficients

Analyzing the Setup

The Ratio Strategy

The Algebraic Dance

The Symmetry of Pascal's Triangle

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