Dany jest ciąg geometryczny (an). Oblicz : a) [latex] S_{7} [/latex], jeśli a1=5 i [latex]q=- frac{1}{2} [/latex] b) n, jeśli [latex] a_{1} =- frac{1}{2} [/latex], [latex]q=- frac{1}{2} [/latex] i [latex] S_{n} = -frac{513}{1536} [/latex]

a) S₇ = a₁ * [(1 - q⁷) / (1 - q) = 5 * [1 - (-1/2)⁷] / [1 - (-1/2)] = 5 * [1 - (-1/128)] / (1 + 1/2) ] = 5 * (1 + 1/128) / (3/2) = 5 * 2/3 * 1 1/128 = 10/3 * 129/128 = 1290 / 384 = 215/64 = 3 23/64 b) Sn = a₁ * [(1 - q^n) / (1 - q)] = -1/2 * [(1 - (-1/2)^n] / (1 + 1/2) = -1/2 * [(1 - (-1/2)^n] / (3/2) = -1/2 * 2/3 * [1 - (-1/2)^n] = -1/3 [1 - (-1/2)^n] = -513/1536        I * (-3) 1 - (-1/2)^n = 513 / 512  1 - (-1/2)^n = 1 1/512       I - 1 -(-1/2)^n = 1/512     I *(-1) (-1/2)^n = -1/512 (-1/2)^n = (-1/2)^9 n = 9

Zadanie rozwiążę w załączniku