The two consecutive terms in the expansion of `(1+x)^(24)` whose coefficients are in the ratio 1:4 are

Options

A. (a) 3rd and 4th
B. (b) 4th and 5th
C. (c) 5th and 6th
D. (d) 6th and 7th

Answer

To solve the problem of finding the two consecutive terms in the expansion of \((1+x)^{24}\) whose coefficients are in the ratio \(1:4\), we can follow these steps:

### Step 1: Identify the general term
The general term \(T_{r+1}\) in the expansion of \((1+x)^n\) is given by the formula:
\[ T_{r+1} = \binom{n}{r} A^{n-r} B^r \]
For our case, \(A = 1\), \(B = x\), and \(n = 24\). Thus, the general term becomes:
\[ T_{r+1} = \binom{24}{r} x^r \]

### Step 2: Write the consecutive terms
The two consecutive terms we are interested in are \(T_{r+1}\) and \(T_{r+2}\):
- The \(r+1\)th term is:
\[ T_{r+1} = \binom{24}{r} x^r \]
- The \(r+2\)th term is:
\[ T_{r+2} = \binom{24}{r+1} x^{r+1} \]

### Step 3: Set up the ratio of coefficients
According to the problem, the coefficients of these terms are in the ratio \(1:4\). Therefore, we can write:
\[ \frac{\binom{24}{r}}{\binom{24}{r+1}} = \frac{1}{4} \]

### Step 4: Simplify the ratio
Using the property of binomial coefficients:
\[ \frac{\binom{n}{r}}{\binom{n}{r+1}} = \frac{r+1}{n-r} \]
we can substitute \(n = 24\):
\[ \frac{\binom{24}{r}}{\binom{24}{r+1}} = \frac{r+1}{24 - r} \]
Thus, we have:
\[ \frac{r+1}{24 - r} = \frac{1}{4} \]

### Step 5: Cross-multiply and solve for \(r\)
Cross-multiplying gives:
\[ 4(r + 1) = 1(24 - r) \]
Expanding both sides:
\[ 4r + 4 = 24 - r \]
Combining like terms:
\[ 4r + r = 24 - 4 \]
\[ 5r = 20 \]
Dividing both sides by 5:
\[ r = 4 \]

### Step 6: Identify the terms
Now that we have \(r = 4\), we can find the two consecutive terms:
- The \(r+1\)th term (5th term) is:
\[ T_5 = \binom{24}{4} x^4 \]
- The \(r+2\)th term (6th term) is:
\[ T_6 = \binom{24}{5} x^5 \]

### Conclusion
Thus, the two consecutive terms in the expansion of \((1+x)^{24}\) whose coefficients are in the ratio \(1:4\) are the 5th and 6th terms.

### Final Answer
The answer is option C: 5th and 6th.

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