7. What is the third term in the expansion of (x+5)^4? a. 20x^2 b. 40x^2 c. 60x^2 d. 80x^2

To find the third term in the expansion of \((x + 5)^4\), we can use the Binomial Theorem. The Binomial Theorem states that:

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

In this case, \(a = x\), \(b = 5\), and \(n = 4\).

The general term (k-th term) in the expansion is given by:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

For the third term (\(k = 2\)):

\[
T_3 = \binom{4}{2} x^{4-2} \cdot 5^2
\]

First, calculate the binomial coefficient:

\[
\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]

Now, plug in the values:

\[
T_3 = 6 \cdot x^2 \cdot 25 = 150x^2
\]

However, the options provided do not match \(150x^2\). There seems to be a discrepancy with the problem statement or options. Let me correct it.

Let's recalculate:

For the third term:

\[
T_3 = \binom{4}{2} \cdot x^{4-2} \cdot 5^2 = 6 \cdot x^2 \cdot 25 = 150x^2
\]

If the problem had a mistake in options, the correct coefficient would be 150x^2.

But if using different coefficients, we'd find the correct answer should be based on the corrected interpretation or problem. Given the process, none match perfectly. **No provided option (a-d) matches this outcome**.

You should verify the problem's coefficients or choices.

What is the third term in the expansion of (x+5)^4? a. 20x^2 b. 40x^2 c. 60x^2 d. 80x^2

Learn how to use the Binomial Theorem to find the third term in the expansion of (x + 5)^4. This problem explores key algebraic concepts and the use of binomial coefficients. Suitable for students in Grades 9-12.

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