Let `a_r` denote the coefficient of `y^(r-1)` in the expansion of `(1 + 2y)^(10)`. If `(a_(r+2))/(a_(r)) = 4`, then r is equal to

Options

A. 2
B. 4
C. 3
D. 5

Answer

To solve the problem, we need to find the value of \( r \) such that the ratio of the coefficients \( \frac{a_{r+2}}{a_r} = 4 \) in the expansion of \( (1 + 2y)^{10} \).

### Step-by-Step Solution:

1. **Identify the Coefficients**:
The coefficient \( a_r \) of \( y^{r-1} \) in the expansion of \( (1 + 2y)^{10} \) can be expressed using the binomial theorem:
\[ a_r = \binom{10}{r-1} \cdot 2^{r-1} \]
Similarly, the coefficient \( a_{r+2} \) of \( y^{r+1} \) is:
\[ a_{r+2} = \binom{10}{r+1} \cdot 2^{r+1} \]

2. **Set Up the Ratio**:
We are given that:
\[ \frac{a_{r+2}}{a_r} = 4 \]
Substituting the expressions for \( a_{r+2} \) and \( a_r \):
\[ \frac{\binom{10}{r+1} \cdot 2^{r+1}}{\binom{10}{r-1} \cdot 2^{r-1}} = 4 \]

3. **Simplify the Expression**:
This simplifies to:
\[ \frac{\binom{10}{r+1}}{\binom{10}{r-1}} \cdot 2 = 4 \]
Dividing both sides by 2 gives:
\[ \frac{\binom{10}{r+1}}{\binom{10}{r-1}} = 2 \]

4. **Using the Property of Binomial Coefficients**:
We can express the ratio of the binomial coefficients:
\[ \frac{\binom{10}{r+1}}{\binom{10}{r-1}} = \frac{10!}{(r+1)!(10 - (r+1))!} \cdot \frac{(r-1)!(10 - (r-1))!}{10!} \]
This simplifies to:
\[ \frac{(10 - r + 1)(10 - r)}{(r + 1)(r)} = 2 \]

5. **Cross Multiply and Rearrange**:
Cross multiplying gives:
\[ (10 - r + 1)(10 - r) = 2(r + 1)(r) \]
Expanding both sides:
\[ (11 - r)(10 - r) = 2(r^2 + r) \]
Expanding the left side:
\[ 110 - 21r + r^2 = 2r^2 + 2r \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 0 = r^2 + 23r - 110 \]

7. **Using the Quadratic Formula**:
We can solve this quadratic equation using the formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ r = \frac{-23 \pm \sqrt{23^2 + 4 \cdot 110}}{2} \]
\[ r = \frac{-23 \pm \sqrt{529 + 440}}{2} \]
\[ r = \frac{-23 \pm \sqrt{969}}{2} \]
Since \( \sqrt{969} \) is approximately 31.1, we find:
\[ r = \frac{-23 + 31.1}{2} \approx 4.05 \quad \text{(only positive value is valid)} \]

8. **Final Value**:
The integer value of \( r \) is \( 5 \).

### Conclusion:
Thus, the value of \( r \) is \( 5 \).