Question

Найдите сумму первых восьми членов геометрической прогрессии 2,6,18,...

Answer 1

$2,\; 6,\; 18\; ,\ldots \\\\q=\frac{b_2}{b_1}=\frac{6}{2}=3\; \; \; \left (q=\frac{b_3}{b_2}=\frac{18}{6}=3\right )\\\\S_{8}= \frac{b_1(1-q^8)}{1-q} = \frac{2(1-3^8)}{1-3} =-(1-3^8)=-(1-6561)=6560$

Answer 2

Calculating the Sum of the First Eight Terms of a Geometric Progression

To find the sum of the first eight terms of the geometric progression 2, 6, 18,...

Step 1: Identify the Key Parameters - The first term, \(a_1\), is 2. - The common ratio, \(q\), is 3.

Step 2: Use the Formula for the Sum of the First \(n\) Terms of a Geometric Progression - The formula for the sum of the first \(n\) terms of a geometric progression is given by: \[S_n = a_1 \times \frac{{1 - q^n}}{{1 - q}}\]

Step 3: Calculate the Sum - Substituting the values of \(a_1\), \(q\), and \(n = 8\) into the formula: \[S_8 = 2 \times \frac{{1 - 3^8}}{{1 - 3}}\]

Step 4: Perform the Calculation - Using the given values: \[S_8 = 2 \times \frac{{1 - 6561}}{{1 - 3}}\] \[S_8 = 2 \times \frac{{-6560}}{{-2}}\] \[S_8 = 6560\]

Step 5: Conclusion - Therefore, the sum of the first eight terms of the geometric progression 2, 6, 18,... is 6560.

[[SOURCE #3]]