Find the sum of the following. (1-1/n) + (1-2/n) + (1-3/n) + ....... up to n terms. 

To solve the problem, let's first write the given series more explicitly:

\[
\text{Sum} = \left(1 - \frac{1}{n}\right) + \left(1 - \frac{2}{n}\right) + \left(1 - \frac{3}{n}\right) + \dots + \left(1 - \frac{n}{n}\right)
\]

We can separate the terms in the series:

\[
\text{Sum} = \left(1 + 1 + 1 + \dots + 1\right) - \left(\frac{1}{n} + \frac{2}{n} + \frac{3}{n} + \dots + \frac{n}{n}\right)
\]

There are \(n\) ones in the first part of the series, so their sum is:

\[
\text{Sum of ones} = n \times 1 = n
\]

Now, let's calculate the sum of the second part:

\[
\text{Sum of fractions} = \frac{1}{n} \times (1 + 2 + 3 + \dots + n)
\]

The sum of the first \(n\) natural numbers \(1 + 2 + 3 + \dots + n\) is given by the formula:

\[
\sum_{k=1}^{n} k = \frac{n(n+1)}{2}
\]

Thus, the sum of the fractions becomes:

\[
\text{Sum of fractions} = \frac{1}{n} \times \frac{n(n+1)}{2} = \frac{n+1}{2}
\]

Finally, subtract the sum of the fractions from the sum of the ones:

\[
\text{Sum} = n - \frac{n+1}{2}
\]

To simplify further:

\[
\text{Sum} = \frac{2n - (n + 1)}{2} = \frac{n - 1}{2}
\]

Thus, the sum of the series is:

\[
\boxed{\frac{n(n-1)}{2}}
\]

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

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Here are five related questions:

1. How does the sum of the first \(n\) natural numbers influence the result of this series?
2. What if the series had started from \( \frac{2}{n} \) instead of \( \frac{1}{n} \)?
3. How would you find the sum if the sequence involved a quadratic function instead of a linear one?
4. Can this method be generalized for any arithmetic series of this form?
5. What are the implications of this series in a real-world context, like probability or statistics?

**Tip:** Recognizing patterns in sequences and series can often lead to more general solutions for related problems.

Learn how to calculate the sum of the series (1 - k/n) for n terms with a step-by-step solution. This problem covers arithmetic series, summation formulas, and their applications. Suitable for high school students studying sequences and series in mathematics.

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