Question

Using Pascal's Triangle, what is the third term in the expansion of (a+5)5 when the expanded polynomial is written in standard from?(1 point)

Answer 1

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:

1. Compute the binomial coefficient:

\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

2. Compute \(a^{5-2}\):

\[
a^{5-2} = a^3
\]

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:

\[
\boxed{250a^3}
\]