To evaluate \( \Delta T(n) = \frac{2}{n(n+1)} - 1 \) for \( n = 1024 \), let's proceed step by step.

### Step 1: Substitute \( n = 1024 \) into the equation
\[
\Delta T(1024) = \frac{2}{1024(1024+1)} - 1
\]

### Step 2: Simplify the expression
\[
\Delta T(1024) = \frac{2}{1024 \times 1025} - 1
\]
\[
\Delta T(1024) = \frac{2}{1049600} - 1
\]

### Step 3: Perform the division
\[
\frac{2}{1049600} = 1.905 \times 10^{-6}
\]

### Step 4: Subtract 1
\[
\Delta T(1024) = 1.905 \times 10^{-6} - 1
\]
\[
\Delta T(1024) = -0.999998095
\]

Thus, \( \Delta T(1024) \approx -0.999998 \).

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Would you like more details on any part of this solution?

Here are some related questions to explore:
1. What happens as \( n \to \infty \) for \( \Delta T(n) \)?
2. Can you rewrite \( \frac{2}{n(n+1)} \) in terms of a simpler expression?
3. How would this function behave for smaller values of \( n \)?
4. What applications might this type of function have in real-world scenarios?
5. How would you graph \( \Delta T(n) \) for a range of \( n \) values?

**Tip**: When dealing with fractions involving large numbers, approximations can sometimes help simplify the process!

Explore how to solve ΔT(n) = 2/n(n+1) − 1 for n = 1024 with a detailed step-by-step solution. This problem demonstrates simplifying rational expressions and performing approximations, suitable for high school students in Grades 10-12.

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