Arithmetic Progressions
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 \(n ? \)