Questions: Use the binomial formula to find the coefficient of the q^13 term in the expansion of (q+2p)^14.

Solution Steps

We start with the expression \( (q + 2p)^{14} \) and apply the binomial theorem, which states that \( (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). Here, \( a = q \), \( b = 2p \), and \( n = 14 \).

To find the coefficient of the \( q^{13} \) term, we need to set \( n - k = 13 \). This gives us \( k = 1 \).

Using the binomial coefficient, we calculate: \[ \text{Coefficient} = \binom{14}{1} (2p)^1 \] Calculating \( \binom{14}{1} = 14 \) and \( (2p)^1 = 2p \), we find: \[ \text{Coefficient} = 14 \cdot 2p = 28p \]

Final Answer