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 form?

Answer 1

To find the third term in the expansion of \((a + 5)^5\) using Pascal's Triangle, we start by identifying the coefficients from the triangle for \(n = 5\).

The coefficients from Pascal's Triangle for \(n = 5\) are:
\[
1, 5, 10, 10, 5, 1
\]

In the binomial expansion of \((x + y)^n\), the general term can be represented as:
\[
T_k = \binom{n}{k} x^{n-k} y^k
\]
where \(T_k\) is the \((k+1)\)-th term in the expansion.

In our case, \(n = 5\), \(x = a\), and \(y = 5\). We need to find the third term, which corresponds to \(k = 2\) (since \(k\) starts from 0).

Using the formula:
\[
T_2 = \binom{5}{2} a^{5-2} (5)^2
\]

First, we calculate the binomial coefficient:
\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

Next, we evaluate \(a^{5-2}\):
\[
a^{5-2} = a^3
\]

We also calculate \(5^2\):
\[
5^2 = 25
\]

Now we combine all parts to find the third term:
\[
T_2 = 10 \cdot a^3 \cdot 25 = 250 a^3
\]

Thus, the third term in the expansion of \((a + 5)^5\) is:
\[
\boxed{250a^3}
\]