If there are three successive coefficients in the expansion of `(1+2x)^(n)` which are in the ratio `1:4:10`, then 'n' is equal to :

Answer

To solve the problem, we need to find the value of \( n \) such that three successive coefficients in the expansion of \( (1 + 2x)^n \) are in the ratio \( 1:4:10 \).

### Step 1: Identify the General Term
The general term \( T_r \) in the expansion of \( (1 + 2x)^n \) is given by:
\[ T_r = \binom{n}{r} (2x)^r = \binom{n}{r} 2^r x^r \]
Thus, the coefficients of the terms are:
- Coefficient of \( T_r \): \( \binom{n}{r} 2^r \)
- Coefficient of \( T_{r+1} \): \( \binom{n}{r+1} 2^{r+1} \)
- Coefficient of \( T_{r+2} \): \( \binom{n}{r+2} 2^{r+2} \)

### Step 2: Set Up the Ratios
According to the problem, these coefficients are in the ratio \( 1:4:10 \). Therefore, we can write:
\[ \frac{\binom{n}{r} 2^r}{\binom{n}{r+1} 2^{r+1}} = \frac{1}{4} \]
and
\[ \frac{\binom{n}{r+1} 2^{r+1}}{\binom{n}{r+2} 2^{r+2}} = \frac{4}{10} = \frac{2}{5} \]

### Step 3: Simplify the First Ratio
From the first ratio:
\[ \frac{\binom{n}{r}}{\binom{n}{r+1}} \cdot \frac{1}{2} = \frac{1}{4} \]
This simplifies to:
\[ \frac{\binom{n}{r}}{\binom{n}{r+1}} = \frac{1}{2} \]
Using the identity \( \frac{\binom{n}{r}}{\binom{n}{r+1}} = \frac{n - r}{r + 1} \):
\[ \frac{n - r}{r + 1} = \frac{1}{2} \]
Cross-multiplying gives:
\[ 2(n - r) = r + 1 \implies 2n - 2r = r + 1 \implies 2n = 3r + 1 \tag{1} \]

### Step 4: Simplify the Second Ratio
From the second ratio:
\[ \frac{\binom{n}{r+1}}{\binom{n}{r+2}} \cdot \frac{1}{2} = \frac{2}{5} \]
This simplifies to:
\[ \frac{\binom{n}{r+1}}{\binom{n}{r+2}} = \frac{4}{5} \]
Using the identity \( \frac{\binom{n}{r+1}}{\binom{n}{r+2}} = \frac{n - r - 1}{r + 2} \):
\[ \frac{n - r - 1}{r + 2} = \frac{4}{5} \]
Cross-multiplying gives:
\[ 5(n - r - 1) = 4(r + 2) \implies 5n - 5r - 5 = 4r + 8 \implies 5n = 9r + 13 \tag{2} \]

### Step 5: Solve the Equations
Now we have two equations:
1. \( 2n = 3r + 1 \)
2. \( 5n = 9r + 13 \)

From equation (1), we can express \( n \) in terms of \( r \):
\[ n = \frac{3r + 1}{2} \]
Substituting this into equation (2):
\[ 5\left(\frac{3r + 1}{2}\right) = 9r + 13 \]
Multiplying through by 2 to eliminate the fraction:
\[ 5(3r + 1) = 18r + 26 \]
Expanding gives:
\[ 15r + 5 = 18r + 26 \]
Rearranging gives:
\[ 3r = 21 \implies r = 7 \]

### Step 6: Find \( n \)
Substituting \( r = 7 \) back into the expression for \( n \):
\[ n = \frac{3(7) + 1}{2} = \frac{21 + 1}{2} = \frac{22}{2} = 11 \]

### Final Answer
Thus, the value of \( n \) is:
\[ \boxed{11} \]