If the coefficients of x⁴, x⁵ and x⁶ in the expansion of (1 + x)ⁿ are in the arithmetic progression,

Q: Why can't we just expand the combinations like ${}^nC_4 = \frac{n!}{4!(n-4)!}$ and solve it?

You absolutely could! However, that path leads to a very messy algebraic expression with many factorials. The 'ratio trick' (dividing by the middle term) is a standard, elegant technique in JEE problems to simplify expressions involving consecutive binomial coefficients. It saves time and reduces the chance of calculation errors.

Q: Why do we get two values for $n$ ($7$ and $14$)? Which one is correct?

Both values are mathematically correct for the A.P. condition. However, the question specifically asks for the *maximum* value of $n$. In competitive exams, always re-read the final requirement to ensure you are providing the specific answer requested.

Q: What if the coefficients were not in A.P. but in G.P. (Geometric Progression)?

If they were in G.P., the condition would be $b^2 = ac$ (i.e., $({}^nC_5)^2 = {}^nC_4 \cdot {}^nC_6$). The strategy of using ratios remains the same, but the resulting equation would look different. Always look for the progression type first!

Question

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1 + x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

Options:

(A) 7

(B) 21

(C) 28

(D) 14

Sigma Percentile · Accepted Answer

### The Elegance of Binomial Coefficients

Welcome, aspiring mathematicians. Today, we embark on a journey to decode the elegant structure of binomial coefficients.

Often, students view the Binomial Theorem as a dry collection of formulas, but it is actually a beautiful landscape of symmetry and patterns. Let us explore this problem together.

### Phase 1: Decoding the A.P. Condition

We are given the expansion of $(1+x)^n$. The general term is $T_{r+1} = {}^nC_r x^r$.

Our task is to focus on the coefficients of $x^4, x^5,$ and $x^6$. These are simply ${}^nC_4, {}^nC_5,$ and ${}^nC_6$.

The problem states that these three values are in an Arithmetic Progression (A.P.).

What does it mean for three numbers $a, b, c$ to be in A.P.? It means the difference between consecutive terms is constant, which leads us to the fundamental condition: $2b = a + c$.

Applying this to our coefficients, we get:

$$2 \cdot {}^nC_5 = {}^nC_4 + {}^nC_6$$

### Phase 2: The Ratio Trick

Now, we face a choice. We could expand these combinations into factorials, but that would be a nightmare of algebra.

Instead, let us use a powerful tool in our arsenal: the ratio property. If we divide the entire equation by the middle term, ${}^nC_5$, we get:

$$2 = \frac{{}^nC_4}{{}^nC_5} + \frac{{}^nC_6}{{}^nC_5}$$

This is much cleaner! Now, we recall the standard ratio property of binomial coefficients:

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

For the first term, $\frac{{}^nC_4}{{}^nC_5}$, we take the reciprocal of the property with $r=5$, giving us $\frac{5}{n-4}$.

For the second term, $\frac{{}^nC_6}{{}^nC_5}$, we use the property directly with $r=6$, giving us $\frac{n-5}{6}$.

### Phase 3: The Algebraic Journey

Substituting these back into our equation, we obtain:

$$2 = \frac{5}{n-4} + \frac{n-5}{6}$$

To clear the fractions, we multiply the entire equation by $6(n-4)$:

$$12(n-4) = 30 + (n-5)(n-4)$$

Expanding both sides carefully, we get:

$$12n - 48 = 30 + (n^2 - 9n + 20)$$

$$12n - 48 = n^2 - 9n + 50$$

Rearranging everything to one side, we arrive at a beautiful quadratic equation:

$$n^2 - 21n + 98 = 0$$

### The Final Resolution

We need to factorize this quadratic. We are looking for two numbers that multiply to $98$ and add to $-21$.

Those numbers are $-7$ and $-14$. Thus, the equation becomes:

$$(n-7)(n-14) = 0$$

This gives us two possible values: $n=7$ or $n=14$.

The question asks for the **maximum** value of $n$. Therefore, we select **$14$**.

Remember, the beauty of mathematics lies not just in the final answer, but in the path we take to get there. By using the ratio property, we turned a potentially daunting problem into a simple, solvable quadratic.

Animated Solution for Mathematics - Binomial Theorem: If the coefficients of $x^{4}, x^{5}$ and $x^{6}$ in the expansion of $(1 + x)^{n}$ are in the arithmetic progression, then the maximum value of $n$ is:

Select Answer:

Visualized Solution

Identifying the Coefficients

Applying the A.P. Condition

Simplifying using Ratios

Using the Ratio Property

Setting up the Algebraic Equation

Expanding the Terms

Forming the Quadratic Equation

Solving the Quadratic

Final Conclusion

The Sigma Insight: Properties of Binomial Coefficients

The Elegance of Binomial Coefficients

Phase 1

Phase 2

Phase 3

The Final Resolution

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