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Find the sum of the first 30 terms in this geometric series:

-5-4-(16)/(5)dots
Choose 1 answer:
(A)
-1.24*10^(20)
(B) -24.97
(c) -2.78
(D) -0.04

Find the sum of the first $30$ terms in this geometric series: $\newline$ $-5-4-\frac{16}{5} \ldots$ $\newline$ Choose $1$ answer: $\newline$ (A) $-1.24 \cdot 10^{20}$ $\newline$ (B) $-24$ . $97$ $\newline$ (C) $-2$ . $78$ $\newline$ (D) $-0$ . $04$

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Solve one-step multiplication and division equations with decimals and fractions

Full solution

Q. Find the sum of the first

30

terms in this geometric series:

\newline

-5-4-\frac{16}{5} \ldots

\newline

Choose

1

answer:

\newline

(A)

-1.24 \cdot 10^{20}

\newline

(B)

-24

.

97

\newline

(C)

-2

.

78

\newline

(D)

-0

.

04

Identify first term and common ratio: Identify the first term ( $a_1$ ) and the common ratio ( $r$ ) of the geometric series. $\newline$ The first term $a_1$ is $-5$ . $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = \frac{-4}{-5} = \frac{4}{5}$
Use sum formula for geometric series: Use the formula for the sum of the first $n$ terms of a geometric series: $S_n = a_1 \times (1 - r^n) / (1 - r)$ , where $n$ is the number of terms. $\newline$ Here, $n = 30$ , $a_1 = -5$ , and $r = \frac{4}{5}$ .
Plug values and calculate sum: Plug the values into the formula and calculate the sum. $\newline$ $S_{30} = -5 \times (1 - (\frac{4}{5})^{30}) / (1 - \frac{4}{5})$
Simplify denominator: Simplify the denominator $1 - \frac{4}{5}$ . $\newline$ $1 - \frac{4}{5} = \frac{1}{5}$
Calculate $(4/5)^{30}$ : Calculate $(4/5)^{30}$ . $\newline$ This is a very small number because $4/5$ is less than $1$ and raising it to the $30th$ power will make it much smaller.
Substitute values into formula: Substitute the values into the formula. $S_{30} = -5 \times \left(1 - \left(\frac{4}{5}\right)^{30}\right) / \left(\frac{1}{5}\right)$
Multiply by $5$ to simplify: Multiply both numerator and denominator by $5$ to simplify the fraction. $S_{30} = -5 \times 5 \times \left(1 - \left(\frac{4}{5}\right)^{30}\right)$
Simplify the multiplication: Simplify the multiplication. $S_{30} = -25 \times (1 - (\frac{4}{5})^{30})$
Approximate sum: Since $(\frac{4}{5})^{30}$ is a very small number, we can approximate $1 - (\frac{4}{5})^{30}$ as being very close to $1$ . $S_{30} \approx -25 \times 1$
Calculate approximate sum: Calculate the approximate sum. $S_{30} \approx -25$

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