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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.

Two sequences \( \{ x, a_1, a_2, a_3, y \} \) and \( \{ x, b_1, b_2, b_3, b_4, b_5, y \} \) each form an arithmetic progression. If \( x \neq y \), what is the value of

\[ \frac{a_2 - a_1}{b_5 - b_4} ? \]

For an arithmetic progression \( \{a_n\} \), the following equalities hold: \[ a_3 + a_5 = 36, \quad a_2 a_4 = 180. \]

Find the largest \( n \) such that \( a_n < 100. \)

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