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# Arithmetic Progressions

What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this.

Find the number of strictly increasing arithmetic progressions of length three, with terms from \(1\) to \(1000\), that consist entirely of perfect squares.

**Details and assumptions**

The **length** of an arithmetic progression is the number of terms that it has.

Find the sum of the 10 terms.

150 workers are engaged to complete a job and it is known that if they all work together the job will be completed in a certain number of days. However, after the first day of work, 4 workers resign. After the second day, another 4 resign. This pattern continues until the job is finally completed, 8 days over schedule.

Find the number of days in which the work was completed.

Consider an arithmetic progression with terms \( a_1, a_2, \ldots \) and the sum of the first \(k\) terms is \(S_k \). If

\[ a_{13} = \frac{1}{11} , a_{11} = \frac{1}{13}, \]

then find \( S_{143} \).

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