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?

Answer 1

To find the coefficient of the second term in the expansion of \((a+3)^7\) using the Binomial Theorem, we start by recognizing the general form of the Binomial Theorem:

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

In this case, we can let \(x = a\), \(y = 3\), and \(n = 7\). The expansion of \((a + 3)^7\) will therefore be:

\[
(a + 3)^7 = \sum_{k=0}^{7} \binom{7}{k} a^{7-k} \cdot 3^k
\]

We are interested in the coefficient of the second term in this expansion. The second term corresponds to \(k = 1\) (noting that the first term corresponds to \(k = 0\)).

To compute the coefficient of the second term, we calculate:

\[
\text{Coefficient} = \binom{7}{1} \cdot 3^1
\]

We first evaluate \(\binom{7}{1}\):

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

Next, we simplify \(3^1\):

\[
3^1 = 3
\]

Now, we multiply these results to find the coefficient of the second term:

\[
\text{Coefficient} = 7 \cdot 3 = 21
\]

Thus, the coefficient of the second term in the expanded form of \((a + 3)^7\) is \(\boxed{21}\).