To find the third term in the expansion of \((a + 5)^5\) using Pascal's Triangle, we can use the binomial theorem. The binomial theorem states that:
\[ (x+y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
In our case, \(x = a\), \(y = 5\), and \(n = 5\).
We will use the coefficients from Pascal's Triangle for \(n = 5\):
- The coefficients are: \(1, 5, 10, 10, 5, 1\).
The general term in the binomial expansion is given by:
\[ T_{k+1} = \binom{n}{k} x^{n-k} y^k \]
We are looking for the third term, which corresponds to \(k = 2\):
\[ T_{3} = \binom{5}{2} a^{5-2} (5)^2 \]
Now we calculate each part:
- Compute the binomial coefficient:
\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]
- Compute \(a^{5-2}\):
\[ a^{5-2} = a^3 \]
- Compute \(5^2\):
\[ 5^2 = 25 \]
Now, we combine these to find the third term:
\[ T_{3} = 10 \cdot a^3 \cdot 25 = 250a^3 \]
Thus, the third term in the expansion of \((a + 5)^5\) is:
\[ \boxed{250a^3} \]
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