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)

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 general term in the expansion can be given as:

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

The second term corresponds to \(k = 1\) (since \(k\) starts from 0). Thus, we will calculate the term when \(k = 1\):

1. Compute the binomial coefficient \(\binom{7}{1}\):

\[
\binom{7}{1} = 7
\]

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

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

Putting this together, the second term in the expansion is:

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

Thus, the coefficient of the second term in the expanded form of \((a + 3)^7\) is:

\[
\boxed{21}
\]