Algebra
# Arithmetic Progressions

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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