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

Explanation

To find the two successive 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 terms
In the expansion of \( (1+x)^{24} \), the general term \( T_k \) is given by:
\[ T_k = \binom{24}{k} x^k \]
where \( k \) is the term number starting from 0. Thus, the coefficients of the terms are \( \binom{24}{r} \) for the \( (r+1) \)-th term and \( \binom{24}{r+1} \) for the \( (r+2) \)-th term.

### Step 2: Set up the ratio
We know that the coefficients of the two successive terms are in the ratio \( 1:4 \). Therefore, we can write:
\[ \frac{\binom{24}{r}}{\binom{24}{r+1}} = \frac{1}{4} \]

### Step 3: Simplify the ratio
Using the property of binomial coefficients:
\[ \frac{\binom{n}{k}}{\binom{n}{k+1}} = \frac{k+1}{n-k} \]
we can rewrite the ratio as:
\[ \frac{r+1}{24-r} = \frac{1}{4} \]

### Step 4: Cross-multiply to solve for \( r \)
Cross-multiplying gives:
\[ 4(r + 1) = 24 - r \]
Expanding this, we get:
\[ 4r + 4 = 24 - r \]

### Step 5: Rearranging the equation
Rearranging the equation yields:
\[ 4r + r = 24 - 4 \]
\[ 5r = 20 \]

### Step 6: Solve for \( r \)
Dividing both sides by 5 gives:
\[ r = 4 \]

### Step 7: Identify the terms
Now that we have \( r = 4 \), we can find the two successive terms:
- The \( (r+1) \)-th term (5th term) is:
\[ T_5 = \binom{24}{4} x^4 \]
- The \( (r+2) \)-th term (6th term) is:
\[ T_6 = \binom{24}{5} x^5 \]

### Step 8: Final answer
Thus, the two successive terms in the expansion of \( (1+x)^{24} \) whose coefficients are in the ratio \( 1:4 \) are:
\[ \binom{24}{4} x^4 \quad \text{and} \quad \binom{24}{5} x^5 \]

---