If the co-efficient of x⁷ and x⁸ in the expansion of (2 + x/3)ⁿ are equal, then the value of n is eq

Q: Why do we use $T_{r+1}$ instead of $T_r$?

The index $r$ starts from 0. So, the first term is $T_1$ (where $r=0$), the second is $T_2$ (where $r=1$), and so on. For $x^7$, we need the power of $b$ to be 7, which corresponds to $r=7$, giving us the 8th term, $T_8$.

Q: Can I just expand the binomial coefficients using factorials immediately?

You could, but it is a trap! It leads to massive expressions that are prone to calculation errors. Always simplify the ratio of coefficients first using the property $\frac{{}^nC_r}{{}^nC_{r+1}} = \frac{r+1}{n-r}$ to keep your algebra clean.

Q: Why does the power of 2 and 3 change when we move them across the equals sign?

When you divide by $2^{n-8}$ or $3^8$, you are applying the laws of exponents: $a^m / a^n = a^{m-n}$. It is just bookkeeping—keep your bases together to make the cancellation obvious.

Question

If the co-efficient of $x^7$ and $x^8$ in the expansion of $(2 + \frac{x}{3})^n$ are equal, then the value of $n$ is equal to ___

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The expression given is $(2 + \frac{x}{3})^n$. We aim to find the value of $n$ such that the coefficients of $x^7$ and $x^8$ are identical.

### The General Term: Your Master Key

To find any specific coefficient, we utilize the general term formula for the expansion $(a+b)^n$, which is $T_{r+1} = {}^nC_r a^{n-r} b^r$.

In this specific case, we have $a=2$ and $b=\frac{x}{3}$. Substituting these values, the general term becomes:

$$T_{r+1} = {}^nC_r (2)^{n-r} \left( \frac{x}{3} \right)^r$$

By isolating the variable part, we identify the coefficient of $x^r$ as:

$$C_r = {}^nC_r \cdot 2^{n-r} \cdot \left( \frac{1}{3} \right)^r$$

### The Setup: Equating the Coefficients

The problem states that the coefficient of $x^7$ is equal to the coefficient of $x^8$. We define these coefficients as $C_7$ and $C_8$ respectively:

$$C_7 = {}^nC_7 \cdot 2^{n-7} \cdot \left( \frac{1}{3} \right)^7$$

$$C_8 = {}^nC_8 \cdot 2^{n-8} \cdot \left( \frac{1}{3} \right)^8$$

Setting these two expressions equal, we obtain the following equation:

$${}^nC_7 \cdot 2^{n-7} \cdot \left( \frac{1}{3} \right)^7 = {}^nC_8 \cdot 2^{n-8} \cdot \left( \frac{1}{3} \right)^8$$

### The Algebraic Dance: Simplification

To isolate $n$, we rearrange the terms to group the binomial coefficients on one side and the constants on the other:

$$\frac{{}^nC_7}{{}^nC_8} = \frac{2^{n-8} \cdot (\frac{1}{3})^8}{2^{n-7} \cdot (\frac{1}{3})^7}$$

Applying the laws of exponents to the right side, we simplify the expression:

$$\frac{2^{n-8}}{2^{n-7}} \cdot \frac{(1/3)^8}{(1/3)^7} = 2^{-1} \cdot \left( \frac{1}{3} \right)^1 = \frac{1}{2} \cdot \frac{1}{3} = \frac{1}{6}$$

For the left side, we use the identity $\frac{{}^nC_r}{{}^nC_{r+1}} = \frac{r+1}{n-r}$. Substituting $r=7$:

$$\frac{{}^nC_7}{{}^nC_8} = \frac{7+1}{n-7} = \frac{8}{n-7}$$

### The Final Reveal

Equating the results from both sides, we arrive at the linear equation:

$$\frac{8}{n-7} = \frac{1}{6}$$

Solving for $n$:

$$n - 7 = 48$$

$$n = 55$$

The value of $n$ that makes the coefficients of $x^7$ and $x^8$ identical is **$n = 55$**.

Animated Solution for Mathematics - Binomial Theorem: If the co-efficient of $x^{7}$ and $x^{8}$ in the expansion of $(2 + \frac{x}{3})^{n}$ are equal, then the value of $n$ is equal to ___

Enter Numerical Value:

Visualized Solution

The Expansion $(2 + \frac{x}{3})^{n}$

General Term $T_{r + 1}$

Coefficient of $x^{7}$

Coefficient of $x^{8}$

Equating the Coefficients

Expanding $^{n} C_{r}$

Simplifying Powers of $2$ and $3$

Simplifying Factorials

Solving for $n$

Final Result $n = 55$

The Sigma Insight: General Term and Middle Term

Analyzing the Setup

The General Term

The Setup

The Algebraic Dance

The Final Reveal

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Animated Solution for Mathematics - Binomial Theorem: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the co-efficient of x7 and x8 in the expansion of (2+3x​)n are equal, then the value of n is equal to ___

Enter Numerical Value:

Visualized Solution

The Expansion .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } (2+3x​)n

General Term .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tr+1​

Coefficient of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x7

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Equating the Coefficients

Expanding .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } nCr​

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