Question

Let the coefficients of three consecutive terms in the binomial expansion of (1 + 2x)n be the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of these three terms, is _______.

Answer 1

Correct Answer: 1120

The general term in the binomial expansion of \( (1 + 2x)^n \) is given by: \[ T_k = \binom{n}{k} (2x)^k = \binom{n}{k} 2^k x^k. \] Thus, the coefficient of \( x^k \) is \( \binom{n}{k} 2^k \). 

Step 1: Expressing the ratio of the coefficients. Let the three consecutive terms be \( T_r, T_{r+1}, T_{r+2} \), and their coefficients be \( \binom{n}{r} 2^r, \binom{n}{r+1} 2^{r+1}, \binom{n}{r+2} 2^{r+2} \), respectively. The ratio of these coefficients is given by: \[ \frac{\binom{n}{r+1} 2^{r+1}}{\binom{n}{r} 2^r} = \frac{5}{2}, \quad \frac{\binom{n}{r+2} 2^{r+2}}{\binom{n}{r+1} 2^{r+1}} = \frac{8}{5}. \] 

Step 2: Simplifying the ratios. Simplifying the first ratio: \[ \frac{\binom{n}{r+1}}{\binom{n}{r}} \cdot 2 = \frac{5}{2} \quad \Rightarrow \quad \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{5}{4}. \] Using the property of binomial coefficients \( \frac{\binom{n}{r+1}}{\binom{n}{r}} = \frac{n-r}{r+1} \), we get: \[ \frac{n-r}{r+1} = \frac{5}{4} \quad \Rightarrow \quad 4(n-r) = 5(r+1) \quad \Rightarrow \quad 4n - 4r = 5r + 5 \quad \Rightarrow \quad 4n = 9r + 5. \] Simplifying the second ratio: \[ \frac{\binom{n}{r+2}}{\binom{n}{r+1}} \cdot 2 = \frac{8}{5} \quad \Rightarrow \quad \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{4}{5}. \] Using the property \( \frac{\binom{n}{r+2}}{\binom{n}{r+1}} = \frac{n-r-1}{r+2} \), we get: \[ \frac{n-r-1}{r+2} = \frac{4}{5} \quad \Rightarrow \quad 5(n-r-1) = 4(r+2) \quad \Rightarrow \quad 5n - 5r - 5 = 4r + 8 \quad \Rightarrow \quad 5n = 9r + 13. \] 

Step 3: Solving the system of equations. We now have the system of equations: \[ 4n = 9r + 5 \quad \text{and} \quad 5n = 9r + 13. \] Subtract the first equation from the second: \[ 5n - 4n = (9r + 13) - (9r + 5) \quad \Rightarrow \quad n = 8. \] 

Step 4: Finding the middle term's coefficient. Substitute \( n = 8 \) into the expression for the coefficient of the middle term: \[ T_{r+1} = \binom{8}{r+1} 2^{r+1}. \] Since \( r = 3 \) (from solving the system of equations), we substitute \( r = 3 \) into the expression for \( T_4 \): \[ T_4 = \binom{8}{4} 2^4 = \binom{8}{4} \cdot 16 = 70 \cdot 16 = 1120. \]