The coefficients fo `x^(n)` in the expansion of `(1+x)^(2n)` and `(1+x)^(2n-1)` are in the ratio

Options

A. `1:2`
B. `1:3`
C. `3:1`
D. `2:1`

Answer

To find the ratio of the coefficients of \( x^n \) in the expansions of \( (1+x)^{2n} \) and \( (1+x)^{2n-1} \), we can follow these steps:

### Step 1: Find the coefficient of \( x^n \) in \( (1+x)^{2n} \)

The general term in the binomial expansion of \( (1+x)^{2n} \) is given by:

\[ T_k = \binom{2n}{k} x^k \]

To find the coefficient of \( x^n \), we set \( k = n \):

\[ \text{Coefficient of } x^n \text{ in } (1+x)^{2n} = \binom{2n}{n} \]

### Step 2: Find the coefficient of \( x^n \) in \( (1+x)^{2n-1} \)

Similarly, the general term in the binomial expansion of \( (1+x)^{2n-1} \) is:

\[ T_k = \binom{2n-1}{k} x^k \]

Again, to find the coefficient of \( x^n \), we set \( k = n \):

\[ \text{Coefficient of } x^n \text{ in } (1+x)^{2n-1} = \binom{2n-1}{n} \]

### Step 3: Form the ratio of the coefficients

Now we can form the ratio of the coefficients we found:

\[ \text{Ratio} = \frac{\text{Coefficient of } x^n \text{ in } (1+x)^{2n}}{\text{Coefficient of } x^n \text{ in } (1+x)^{2n-1}} = \frac{\binom{2n}{n}}{\binom{2n-1}{n}} \]

### Step 4: Simplify the ratio

Using the property of binomial coefficients, we have:

\[ \binom{2n}{n} = \frac{(2n)!}{n!n!} \]
\[ \binom{2n-1}{n} = \frac{(2n-1)!}{n!(n-1)!} \]

Thus, the ratio becomes:

\[ \text{Ratio} = \frac{\frac{(2n)!}{n!n!}}{\frac{(2n-1)!}{n!(n-1)!}} = \frac{(2n)!}{(2n-1)!} \cdot \frac{(n-1)!}{n!} \]

Now, simplifying this:

\[ \text{Ratio} = \frac{(2n)(2n-1)!}{(2n-1)!} \cdot \frac{(n-1)!}{n!} = \frac{2n}{n} = 2 \]

### Final Result

Thus, the ratio of the coefficients of \( x^n \) in the expansions of \( (1+x)^{2n} \) and \( (1+x)^{2n-1} \) is:

\[ \text{Ratio} = 2:1 \]

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