To find the first three terms of the binomial expansion of \((m^5 + 5)^{97}\), we will use the **Binomial Theorem**. This theorem provides a way to expand expressions of the form \((a + b)^n\) as a sum of terms involving binomial coefficients.

### Binomial Theorem

The Binomial Theorem states:

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

For the binomial \((m^5 + 5)^{97}\), we have \(a = m^5\), \(b = 5\), and \(n = 97\).

The general term in the expansion is given by:

\[
T_{k+1} = \binom{n}{k} (m^5)^{n-k} (5)^k
\]

where \(T_{k+1}\) is the \((k+1)\)-th term, \(\binom{n}{k}\) is the binomial coefficient, and \(k = 0, 1, 2, \ldots, n\).

### First 3 Terms of the Expansion:

1. **First term (\(T_1\))**: \(k = 0\)

\[
T_1 = \binom{97}{0} (m^5)^{97} (5)^0 = 1 \cdot (m^5)^{97} \cdot 1 = m^{485}
\]

2. **Second term (\(T_2\))**: \(k = 1\)

\[
T_2 = \binom{97}{1} (m^5)^{96} (5)^1 = 97 \cdot (m^5)^{96} \cdot 5 = 485 \cdot m^{480}
\]

3. **Third term (\(T_3\))**: \(k = 2\)

\[
T_3 = \binom{97}{2} (m^5)^{95} (5)^2 = \frac{97 \cdot 96}{2} \cdot (m^5)^{95} \cdot 25 = 4656 \cdot m^{475}
\]

### Simplifying These Terms:

1. \(m^{485}\)
2. \(485 \cdot m^{480}\)
3. \(4656 \cdot m^{475}\)

### Conclusion:

The first three terms of the binomial expansion of \((m^5 + 5)^{97}\) are:

1. \(m^{485}\)
2. \(485 \cdot m^{480}\)
3. \(4656 \cdot m^{475}\)

Would you like to see further details or have any questions?

Here are 5 related questions to explore further:
1. How does the Binomial Theorem apply when there is a negative term inside the binomial?
2. What would be the fourth and fifth terms of this expansion?
3. How can binomial expansions be used in probability theory?
4. What is the significance of the binomial coefficients in the expansion?
5. How would you use the binomial expansion to estimate values for large exponents?

**Tip:** Binomial expansions can be very helpful for approximations when \(n\) is large, especially when focusing on the first few terms of the expansion.

Explore the first three terms of the binomial expansion (m^5 + 5)^97. Understand the application of the Binomial Theorem, including detailed calculations and simplification steps. Suitable for advanced mathematics students.

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