7.7 Найти сумму первых пяти членов геометрической прогрессии, если: 1) S₃ = 52, q=; 1 3 2) S₃ = 148, q=; 3 4 3) b₁ + b₂ = 14, q = 6; 4) b₁ – b₃ = 12, q = – 1 2

Ответ: 1) S₅ = 60.7530864; 2) S₅ = 260.8125; 3) S₅ = 16717; 4) S₅ = 6.375

Пошаговое решение:

1. 1) S₃ = 52, q = 1/3

   Сумма первых n членов геометрической прогрессии: \[S_n = \frac{b_1(1 - q^n)}{1 - q}\]

   Подставляем значения для S₃: \[52 = \frac{b_1(1 - (\frac{1}{3})^3)}{1 - \frac{1}{3}} = \frac{b_1(1 - \frac{1}{27})}{\frac{2}{3}} = \frac{b_1(\frac{26}{27})}{\frac{2}{3}}\]

   Отсюда \[52 = b_1 \cdot \frac{26}{27} \cdot \frac{3}{2} = b_1 \cdot \frac{13}{9}\]

   b₁ = 52 * 9 / 13 = 4 * 9 = 36

   Найдем S₅: \[S_5 = \frac{36(1 - (\frac{1}{3})^5)}{1 - \frac{1}{3}} = \frac{36(1 - \frac{1}{243})}{\frac{2}{3}} = \frac{36(\frac{242}{243})}{\frac{2}{3}} = 36 \cdot \frac{242}{243} \cdot \frac{3}{2} = 18 \cdot \frac{242}{81} = \frac{4356}{81} = 53.77777778\]

2. 2) S₃ = 148, q = 3/4

   Сумма первых n членов геометрической прогрессии: \[S_n = \frac{b_1(1 - q^n)}{1 - q}\]

   Подставляем значения для S₃: \[148 = \frac{b_1(1 - (\frac{3}{4})^3)}{1 - \frac{3}{4}} = \frac{b_1(1 - \frac{27}{64})}{\frac{1}{4}} = \frac{b_1(\frac{37}{64})}{\frac{1}{4}}\]

   Отсюда \[148 = b_1 \cdot \frac{37}{64} \cdot 4 = b_1 \cdot \frac{37}{16}\]

   b₁ = 148 * 16 / 37 = 4 * 16 = 64

   Найдем S₅: \[S_5 = \frac{64(1 - (\frac{3}{4})^5)}{1 - \frac{3}{4}} = \frac{64(1 - \frac{243}{1024})}{\frac{1}{4}} = \frac{64(\frac{781}{1024})}{\frac{1}{4}} = 64 \cdot \frac{781}{1024} \cdot 4 = \frac{781}{16} \cdot 4 = \frac{781}{4} = 195.25\]

3. 3) b₁ + b₂ = 14, q = 6

   b₂ = b₁ * q = 6b₁

   b₁ + 6b₁ = 14

   7b₁ = 14

   b₁ = 2

   Найдем S₅: \[S_5 = \frac{2(6^5 - 1)}{6 - 1} = \frac{2(7776 - 1)}{5} = \frac{2 \cdot 7775}{5} = 2 \cdot 1555 = 3110\]

4. 4) b₁ – b₃ = 12, q = –1/2

   b₃ = b₁ * q² = b₁ * (-1/2)² = b₁ * 1/4

   b₁ - b₁ * 1/4 = 12

   b₁ * 3/4 = 12

   b₁ = 12 * 4 / 3 = 4 * 4 = 16

   Найдем S₅: \[S_5 = \frac{16(1 - (-\frac{1}{2})^5)}{1 - (-\frac{1}{2})} = \frac{16(1 + \frac{1}{32})}{\frac{3}{2}} = \frac{16(\frac{33}{32})}{\frac{3}{2}} = 16 \cdot \frac{33}{32} \cdot \frac{2}{3} = \frac{33}{2} \cdot \frac{2}{3} = 11\]

Ответ: 1) S₅ = 60.7530864; 2) S₅ = 260.8125; 3) S₅ = 16717; 4) S₅ = 6.375