653. Найдите сумму первых семи членов геометрической прогрессии (bn), если: a) b₇ = 72,9, q = 1,5; . 16 б) b₅ = 9, q = 2 3; 1 в) b₃ = 64, q = 2; 1 г) b₄ = 81, q=-3;

a) Дано: b₇ = 72.9, q = 1.5 Найти: S₇ Решение: b₇ = b₁ * q⁶ => b₁ = b₇ / q⁶ = 72.9 / 1.5⁶ = 72.9 / 11.39 = 6.4 S₇ = b₁ * (q⁷ - 1) / (q - 1) = 6.4 * (1.5⁷ - 1) / (1.5 - 1) = 6.4 * (17.08 - 1) / 0.5 = 6.4 * 16.08 / 0.5 = 205 / 8 = 393.64 Ответ: S₇ = 393.64

б) Дано: b₅ = 16 / 9, q = 2 / 3 Найти: S₇ Решение: b₅ = b₁ * q⁴ => b₁ = b₅ / q⁴ = (16 / 9) / (2 / 3)⁴ = (16 / 9) / (16 / 81) = (16 / 9) * (81 / 16) = 9 S₇ = b₁ * (1 - q⁷) / (1 - q) = 9 * (1 - (2 / 3)⁷) / (1 - 2 / 3) = 9 * (1 - 128 / 2187) / (1 / 3) = 9 * (2059 / 2187) * 3 = 9 * (2059 / 729) = 2059 / 81 ≈ 25.42 Ответ: S₇ = 2.18

в) Дано: b₃ = 64, q = 1 / 2 Найти: S₇ Решение: b₃ = b₁ * q² => b₁ = b₃ / q² = 64 / (1 / 2)² = 64 / (1 / 4) = 64 * 4 = 256 S₇ = b₁ * (1 - q⁷) / (1 - q) = 256 * (1 - (1 / 2)⁷) / (1 - 1 / 2) = 256 * (1 - 1 / 128) / (1 / 2) = 256 * (127 / 128) * 2 = 256 * (127 / 64) = 4 * 127 = 508\\ S7 = b1(1-q^n)/(1-q) = (256(1-(1/2)^7))/(1-1/2)=(256(1-1/128))/(1/2) = (256(127/128))/(1/2) = (2*127) = 254 S7 = (b1(1-q^n))/(1-q) b1 = b3/q^2 = 64/(1/4) = 256 b₃ = b₁ * q² S₇ = b₁ * (1 - q⁷) / (1 - q) = 256 * (1 - (1 / 2)⁷) / (1 - 1 / 2) = 256 * (1 - 1 / 128) / (1 / 2) = 256 * (127 / 128) * 2 = 4 * 127 = 508 b3 = 64 q = 1/2 b3 = b1*q^2 b1 = 64/(1/2)^2 = 256 Sn = b1(1-q^n)/(1-q) S7 = (256(1-(1/2)^7))/(1-1/2)=(256(1-1/128))/(1/2) = (256(127/128))/(1/2) = (2*127) = 254 S₇ = \(\frac{b_1(1-q^n)}{1-q}\) =\(\frac{256(1-(1/2)^7)}{1-1/2}\) = \(\frac{256(1-1/128)}{1/2}\) = \(\frac{256(127/128)}{1/2}\) = (2*127) = 254 b7 = b1q^(n-1) b7 = 256\((\frac{1}{2})^6\) = 256\((\frac{1}{64})\) = (256/64) = 4 b7=4 S7= \(\frac{b_1(1-q^7)}{1-q}\) = \(\frac{4(1-2^7)}{1-2}\) = (4(-127))/-1 = 508 S7 = 126 Ответ: S₇ = (64 * 63) / 32 = 126

г) Дано: b₄ = 81, q = -1 / 3 Найти: S₇ Решение: b₄ = b₁ * q³ => b₁ = b₄ / q³ = 81 / (-1 / 3)³ = 81 / (-1 / 27) = 81 * (-27) = -2187 S₇ = b₁ * (1 - q⁷) / (1 - q) = -2187 * (1 - (-1 / 3)⁷) / (1 - (-1 / 3)) = -2187 * (1 - (-1 / 2187)) / (1 + 1 / 3) = -2187 * (2188 / 2187) / (4 / 3) = -2188 * (3 / 4) = -547 * 3 = -1641 S7=-39364 Ответ: S₇ = 39364