Question

Знайдіть суму перших шести членів геометричної прогресії 72. -36. 18...

Answer 1

Ответ:

47,25 - сумма шести первых членов геометрической прогрессии.

Пошаговое объяснение:

По условию задана геометрическая прогрессия 72; -36; 18; ...

Тогда первый член равен 72, второй член равен -36. Найдем знаменатель геометрической прогрессии. Для этого второй член разделим на первый.

$b{_1}= 72;\\b{_2}= -36;\\\\q=\dfrac{b{_2}}{b{_1}} ;\\\\q=\dfrac{-36}{72}=-\dfrac{1}{2}$

Найдем сумму шести первых членов геометрической прогрессии по формуле суммы n- первых членов

$S{_n}= \dfrac{b{_1}(q^{n}-1 )}{q-1} ;\\\\S{_6}= \dfrac{b{_1}(q^{6}-1 )}{q-1} ;$

Тогда получим

$S{_6}= \dfrac{72\cdot \left(\left(-\dfrac{1}{2}\right) ^{6}-1 \right)}{-\dfrac{1}{2} -1}= \dfrac{72\cdot \left(\left\dfrac{1}{64}-1 \right)}{-\dfrac{3}{2} }=72\cdot \left(\left\dfrac{1}{64}-\dfrac{64}{64} \right)\cdot\left(-\dfrac{2}{3}\right )=$

$=72\cdot \left(-\dfrac{63}{64}\right )\cdot\left(-\dfrac{2}{3}\right )=\dfrac{72\cdot63\cdot2}{64\cdot3} =\dfrac{72\cdot21}{32} =\dfrac{9\cdot21}{4} =\dfrac{189}{4} =47\dfrac{1}{4} =47,25$

Answer 2

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

To find the sum of the first six terms of a geometric progression with the given terms, we can use the formula for the sum of the first n terms of a geometric progression:

Sum of the first n terms of a geometric progression: \[ S_n = \frac{a(1 - r^n)}{1 - r} \]

Where: - \( S_n \) = Sum of the first n terms - \( a \) = First term - \( r \) = Common ratio - \( n \) = Number of terms

Applying the Formula

Given the terms of the geometric progression: - First term (\( a \)) = 72 - Common ratio (\( r \)) = -2 (obtained by dividing each term by its preceding term: -36 ÷ 72 = -0.5, 18 ÷ -36 = -0.5)

We want to find the sum of the first six terms (\( n = 6 \)).

Using the formula, we can calculate the sum: \[ S_6 = \frac{72(1 - (-2)^6)}{1 - (-2)} \]

\[ S_6 = \frac{72(1 - 64)}{1 + 2} \]

\[ S_6 = \frac{72(-63)}{3} \]

\[ S_6 = -1512 \]

Answer

The sum of the first six terms of the given geometric progression is -1512.