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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. See more

Problem Solving

Non-zero numbers \( a, b, c, d, e \) form an arithmetic progression. If \[ \frac{b+d}{2} + \frac{a+e}{4} = kc, \] find the value of \( k \).

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 \( a_n < 100. \)

An arithmetic progression \( \{a_n \} \) satisfies \[ a_2 + a_4 + a_6 + \cdots + a_{2n} = 2n^2 + 3n. \] Find the value of \[ a_3 + a_6 + a_9 + \cdots + a_{18}. \]

For an arithmetic progression \( \{ a_n \} \), \( a_3 = 8\) and \(a_{10} = 29 \). If \( a_n = 200 \), what is \(n ? \)

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