The sum of the coefficients of three consecutive terms in the binomial expansion of (1 + x)ⁿ⁺², whic

Q: Wait, why can't we just expand $(1 + x)^{n+2}$ and find the coefficients?

Expanding $(1 + x)^{n+2}$ would result in $n+3$ terms. Without knowing $n$, you would be left with an algebraic nightmare of unknown variables. The ratio property is an elegant shortcut designed specifically to bypass the expansion and isolate the variables $n$ and $r$ directly.

Q: How do I know which term is $r$ and which is $r-1$?

The labeling is arbitrary, but consistency is vital. If you define the middle term as $^{n+2}C_r$, then the term before it must be $^{n+2}C_{r-1}$ and the term after it must be $^{n+2}C_{r+1}$. As long as you maintain this relative order, the math will resolve perfectly.

Q: Is there a trick to remembering the ratio formula?

Think of it as a balance. The numerator $(n-r+1)$ and the denominator $r$ are derived from the factorial definition of combinations. Just remember: the 'top' index stays constant, while the 'bottom' index shifts by one. It is a beautiful piece of symmetry that appears everywhere in combinatorics.

Question

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n+2}$, which are in the ratio $1 : 3 : 5$, is equal to

Options:

(A) 92

(B) 63

(C) 41

(D) 25

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the edge of a vast mathematical landscape, looking at the expansion of $(1 + x)^{n+2}$. It looks daunting, but you do not need to expand it or write out every single term.

Mathematics, at its highest level, is about finding the hidden structures that allow us to bypass brute force. In this problem, we are given three consecutive coefficients in the ratio $1 : 3 : 5$. This ratio is a geometric constraint that forces the variables $n$ and $r$ into a specific, solvable configuration.

### The Mathematical Sword: The Ratio Formula

When we talk about binomial coefficients, we are talking about the entries in Pascal's Triangle. The relationship between any two adjacent entries is governed by the formula:

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

This is your sword. It is the tool that cuts through the complexity. In our case, the index is $n+2$, so we adapt our tool accordingly:

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

### The System of Equations

We define our three terms as $C_1 = ^{n+2}C_{r-1}$, $C_2 = ^{n+2}C_r$, and $C_3 = ^{n+2}C_{r+1}$. Given the ratio $C_1 : C_2 : C_3 = 1 : 3 : 5$, we derive two powerful equations.

First, consider the ratio $\frac{C_2}{C_1} = 3$:

$$\frac{(n+2) - r + 1}{r} = 3 \Rightarrow n + 3 - r = 3r \Rightarrow n + 3 = 4r$$

Next, consider the ratio $\frac{C_3}{C_2} = \frac{5}{3}$:

$$\frac{(n+2) - (r+1) + 1}{r+1} = \frac{5}{3} \Rightarrow \frac{n + 2 - r}{r + 1} = \frac{5}{3}$$

Cross-multiplying the second ratio yields $3(n + 2 - r) = 5(r + 1)$, which simplifies to:

$$3n + 6 - 3r = 5r + 5 \Rightarrow 3n + 1 = 8r$$

### The Resolution

We now have a system of two linear equations: $n + 3 = 4r$ and $3n + 1 = 8r$. To solve this, multiply the first equation by $2$ to align the $r$ terms:

$$2n + 6 = 8r$$

Equating the two expressions for $8r$, we find:

$$3n + 1 = 2n + 6 \Rightarrow n = 5$$

Substituting $n = 5$ back into the first equation, we get $5 + 3 = 4r$, which results in $r = 2$. The index of our expansion is $n + 2 = 7$, and our coefficients are $^7C_1$, $^7C_2$, and $^7C_3$.

Calculating these values:

$$^7C_1 = 7, \quad ^7C_2 = 21, \quad ^7C_3 = 35$$

The final step is to sum these coefficients:

$$7 + 21 + 35 = 63$$

The complexity dissolves, the variables align, and we arrive at the elegant answer of **63**.

Animated Solution for Mathematics - Binomial Theorem: The sum of the coefficients of three consecutive terms in the binomial expansion of $(1 + x)^{n + 2}$ , which are in the ratio $1 : 3 : 5$ , is equal to

Select Answer:

Visualized Solution

Define the Consecutive Terms

Identify the Ratio Formula

Setup the First Ratio Equation

Simplify the First Equation

Setup the Second Ratio Equation

Simplify the Second Equation

Solve for n

Find the value of r

Identify the Three Coefficients

Calculate the Final Sum

Conclusion & Key Takeaway

The Sigma Insight: Properties of Binomial Coefficients

Analyzing the Setup

The Mathematical Sword

The System of Equations

The Resolution

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