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 \(\boxed{21}\).
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