2 Bulgarian Winter Tournament

Question

A_n = 1 2 + 3 4 + 5 8 + + (2n - 1) 2^n. (Nedyalka Dimitrova)

Accepted Answer

Since 2A_n = 1 4 + 3 8 + + (2n - 3) 2^n + (2n - 1) 2^n+1, it follows align* A_n = 2A_n - A_n &= (2n - 1) 2^n+1 - (1 2 + 2 4 + 2 8 + + 2 2^n) &= (2n - 1) 2^n+1 - 2 (2 + 4 + 8 + + 2^n) + 1 2 &= (2n - 1) 2^n+1 - 2 2 2^n - 12 - 1 + 2 = (2n - 3) 2^n+1 + 6. align* Therefore A_2024 = 4045 2^2025 + 6.