Prove that if T(n) = 15n3
 + n2
 + 4, T(n) = O(n4
). 

To prove that \( T(n) = 15n^3 + n^2 + 4 \) is \( O(n^4) \), we need to show that there exist constants \( C \) and \( n_0 \) such that:

\[
T(n) \leq C \cdot n^4 \quad \text{for all} \quad n \geq n_0.
\]

### Step-by-Step Proof:

1. **Expression for \( T(n) \)**:
   \[
   T(n) = 15n^3 + n^2 + 4
   \]

2. **Set up the inequality**:
   We need to show that there exists a constant \( C \) such that:
   \[
   15n^3 + n^2 + 4 \leq C \cdot n^4.
   \]
   
3. **Find a constant \( C \)**:
   Divide the entire expression by \( n^4 \) to compare the terms:

   \[
   \frac{T(n)}{n^4} = \frac{15n^3}{n^4} + \frac{n^2}{n^4} + \frac{4}{n^4} = \frac{15}{n} + \frac{1}{n^2} + \frac{4}{n^4}.
   \]

   As \( n \to \infty \), all of these terms approach 0. This suggests that for large \( n \), the ratio \( \frac{T(n)}{n^4} \) can be made arbitrarily small by choosing \( n \) large enough.

4. **Bound each term**:
   For large \( n \), we observe that:
   \[
   \frac{15}{n} + \frac{1}{n^2} + \frac{4}{n^4} \leq \frac{16}{n},
   \]
   which tends to 0 as \( n \to \infty \).

5. **Conclusion**:
   Since \( \frac{T(n)}{n^4} \) tends to 0 as \( n \) becomes large, we can choose a constant \( C \) such that:
   \[
   15n^3 + n^2 + 4 \leq C \cdot n^4
   \]
   for all \( n \geq n_0 \). For instance, taking \( C = 16 \) works for large \( n \).

Thus, \( T(n) = 15n^3 + n^2 + 4 \) is \( O(n^4) \).

---

Would you like more details or have any questions about this proof?

Here are five related questions to expand the discussion:
1. What is the formal definition of Big-O notation?
2. How do you prove \( T(n) = O(n^k) \) for a different \( k \)?
3. What is the difference between Big-O and Theta notation?
4. How does Big-O analysis help in algorithm efficiency?
5. Can we determine a tighter bound for \( T(n) \) than \( O(n^4) \)?

**Tip**: Always aim to find the tightest possible Big-O bound for more accurate complexity analysis!

Prove that if T(n) = 15n^3 + n^2 + 4, T(n) = O(n^4).

This page provides a step-by-step proof showing that the function T(n) = 15n^3 + n^2 + 4 is O(n^4). Explore the principles of Big-O notation and polynomial growth, including detailed comparisons of terms and constants. Suitable for undergraduate students studying asymptotic analysis.

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