If some three consecutive in the binomial expansion of (x + 1)ⁿ in powers of x are in the ratio 2 :

Q: Wait, why can't I just expand $(x + 1)^n$ and find the coefficients directly? Isn't that easier?

That is a very natural instinct! However, the problem is that $n$ is an unknown variable. If you try to expand $(x + 1)^n$ without knowing $n$, you are essentially trying to build a house without a blueprint. The binomial theorem gives us the general term, but the ratio property allows us to bypass the expansion entirely and solve for the parameters $n$ and $r$ directly. It is the difference between brute force and surgical precision.

Q: I always get confused with the indices in the ratio formula. Is it $\frac{n-r+1}{r}$ or $\frac{n-r}{r+1}$? How do I remember?

It is a classic point of confusion! Think of it this way: when you move from $^nC_{r-1}$ to $^nC_r$, the index $r$ increases. The formula $\frac{n-r+1}{r}$ is derived from the definition of combinations. A great mnemonic is to look at the 'next' term index. If you are going to the $r$-th term, the denominator is $r$. The numerator is the remaining slots available, which is $n - (r - 1) = n - r + 1$. Keep practicing the derivation, and it will become second nature!

Q: Does the order of the ratio matter? What if the problem gave me $70 : 15 : 2$ instead?

That is an excellent question! The order is absolutely critical. The ratio $2 : 15 : 70$ tells us the progression is increasing. If it were $70 : 15 : 2$, the progression would be decreasing, which would imply we are looking at the latter half of the binomial expansion (where coefficients start to shrink). The math remains the same, but your interpretation of the position of $r$ would shift. Always respect the order given in the problem!

Question

If some three consecutive in the binomial expansion of $(x + 1)^n$ in powers of $x$ are in the ratio $2 : 15 : 70$, then the average of these three coefficients is :-

Options:

(A) 964

(B) 625

(C) 227

(D) 232

Sigma Percentile · Accepted Answer

### The Symphony of Binomial Coefficients

Welcome, future engineer. Today, we are not just solving a math problem; we are uncovering the hidden architecture of the Binomial Theorem.

When you look at the expansion of $(x + 1)^n$, you might see a daunting string of terms. But I want you to see something else: a sequence of numbers that dance in a perfect, predictable rhythm.

We are given three consecutive coefficients in the ratio $2 : 15 : 70$. This isn't just a random set of numbers; it is a snapshot of a specific moment in the expansion. Let us embark on this journey to decode it.

### Phase 1: The Geometry of Ratios

Imagine these coefficients as bars on a graph. We have three consecutive terms: $^nC_{r-1}$, $^nC_r$, and $^nC_{r+1}$.

The problem tells us their ratio is $2 : 15 : 70$. Notice the rapid growth? This tells us we are in the early stages of the expansion, where the coefficients are climbing toward the peak of the binomial distribution.

We do not need to know the value of $n$ yet. We only need the relationship between neighbors. The bridge between these neighbors is the ratio property:

$$
\frac{^nC_r}{^nC_{r-1}} = \frac{n-r+1}{r}
$$

This formula is the heartbeat of binomial problems. It allows us to leapfrog from one term to the next without ever needing to calculate the full factorial values.

### Phase 2: The Algebraic Bridge

Now, let us translate the problem into the language of algebra. We have two pairs of ratios.

First, the ratio of the first two terms: $\frac{^nC_{r-1}}{^nC_r} = \frac{2}{15}$. Using our property, we invert it to get $\frac{^nC_r}{^nC_{r-1}} = \frac{15}{2}$.

Substituting the formula, we get:

$$
\frac{n-r+1}{r} = \frac{15}{2}
$$

Cross-multiplying gives us $2n - 2r + 2 = 15r$, which simplifies beautifully to $17r = 2n + 2$. This is our first anchor point.

Next, we look at the second pair: $\frac{^nC_r}{^nC_{r+1}} = \frac{15}{70}$. Simplifying the fraction, we get $\frac{3}{14}$.

Again, using the ratio property, we know that $\frac{^nC_{r+1}}{^nC_r} = \frac{n-r}{r+1}$. Therefore:

$$
\frac{n-r}{r+1} = \frac{14}{3}
$$

Cross-multiplying here yields $3n - 3r = 14r + 14$, or $3n - 17r = 14$. Do you see the elegance? We have transformed a complex binomial problem into a simple system of two linear equations.

### Phase 3: The Moment of Clarity

We have our system:
1) $17r = 2n + 2$
2) $3n - 17r = 14$

Look at the beauty of the substitution. We can replace $17r$ in the second equation with $(2n + 2)$.

This gives us $3n - (2n + 2) = 14$. Suddenly, the variables collapse. $n - 2 = 14$, which means **$n = 16$**.

The mystery of the expansion is solved! With $n = 16$, we plug it back into our first equation: $17r = 2(16) + 2 = 34$. Thus, **$r = 2$**.

We have found the exact location in the expansion: the 1st, 2nd, and 3rd terms (since $r-1=1$, $r=2$, $r+1=3$).

### The Final Celebration

Now, we calculate the actual coefficients. We are looking for $^{16}C_1$, $^{16}C_2$, and $^{16}C_3$.

$$
^{16}C_1 = 16
$$

$$
^{16}C_2 = \frac{16 	imes 15}{2} = 120
$$

$$
^{16}C_3 = \frac{16 	imes 15 	imes 14}{3 	imes 2 	imes 1} = 560
$$

The average of these three is:

$$
\frac{16 + 120 + 560}{3} = \frac{696}{3} = 232
$$

We have arrived at the answer: **$232$**.

But more importantly, you have mastered the technique. You have seen how to turn a ratio of coefficients into a system of equations. This is the power of the Binomial Theorem—it is not about memorizing formulas, but about understanding the relationships between the numbers. Keep this logic in your toolkit, and no binomial problem will ever intimidate you again.

Animated Solution for Mathematics - Binomial Theorem: If some three consecutive in the binomial expansion of $(x + 1)^{n}$ in powers of $x$ are in the ratio $2 : 15 : 70$ , then the average of these three coefficients is :-

Select Answer:

Visualized Solution

Identify Consecutive Coefficients

Establish the First Ratio

Derive First Equation

Establish the Second Ratio

Derive Second Equation

Solve for n

Solve for r

Calculate the Coefficients

Find the Average

Conclusion & Key Takeaway

The Sigma Insight: Properties of Binomial Coefficients

The Symphony of Binomial Coefficients

Phase 1

Phase 2

Phase 3

The Final Celebration

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