Question

Using Pascal’s Triangle, what is the coefficient of the second term in the expanded form of (a+3)7 when the expanded polynomial is written in standard form?(1 point) Responses 21 21 24 24 7 7 3

Answer 1

To find the coefficient of the second term in the expansion of \((a + 3)^7\), we can use the binomial theorem, which states:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case, \(a = a\), \(b = 3\), and \(n = 7\). The \(k\)-th term in the expansion is given by:

\[
\binom{n}{k} a^{n-k} b^k
\]

Specifically, we're interested in the second term, which corresponds to \(k=1\).

1. Calculate the binomial coefficient \(\binom{7}{1}\):
\[
\binom{7}{1} = 7
\]

2. Calculate \(a^{7-1} = a^6\).

3. Calculate \(b^1 = 3^1 = 3\).

Putting this together, the second term is:

\[
\binom{7}{1} a^{6} \cdot 3^1 = 7 \cdot a^{6} \cdot 3 = 21a^6
\]

Thus, the coefficient of the second term in the expansion is \(21\).

Therefore, the answer is:

**21**.