sum_(k=0)^(19)6(1.5)^(k)~~ Choose 1 answer: (A) 7983.02 (B) 13,301.03 (C) 26,590.05 (D) 39,891.08

Recognize as geometric series: Recognize the series as a geometric series where the first term $ a = 6 $ and the common ratio $ r = 1.5 $. The sum of a finite geometric series can be calculated using the formula $ S_n = a \frac{1 - r^n}{1 - r} $, where $ n $ is the number of terms.Calculate number of terms: Calculate the number of terms $ n $. Since the series starts at $ k=0 $ and goes up to $ k=19 $, there are $ 19 - 0 + 1 = 20 $ terms.Substitute values into formula: Substitute the values into the sum formula for a geometric series. Here, $ a = 6 $, $ r = 1.5 $, and $ n = 20 $. So, $ S_{20} = 6 \frac{1 - (1.5)^{20}}{1 - 1.5} $.Calculate sum: Calculate the sum $ S_{20} $. First, calculate $ (1.5)^{20} $ and then substitute it into the formula.Handle negative denominator: Since $ r > 1 $, the denominator $ 1 - r $ is negative. Therefore, we need to be careful with the signs when calculating the sum. The correct formula application is $ S_{20} = 6 \frac{1 - (1.5)^{20}}{1 - 1.5} = 6 \frac{1 - (1.5)^{20}}{-0.5} $.Compute (1.5)^20: Compute $ (1.5)^{20} $ using a calculator. $ (1.5)^{20} \approx 33219.476736 $Substitute into formula and calculate: Substitute $ (1.5)^{20} $ into the sum formula and calculate $ S_{20} $. $ S_{20} = 6 \frac{1 - 33219.476736}{-0.5} $ $ S_{20} = 6 \frac{-33218.476736}{-0.5} $ $ S_{20} = 6 \cdot 66436.953472 $ $ S_{20} = 398621.720832 $

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sum_(k=0)^(19)6(1.5)^(k)~~
Choose 1 answer:
(A) 7983.02
(B)
13,301.03
(C)
26,590.05
(D)
39,891.08

$\sum_{k=0}^{19} 6(1.5)^{k} \approx$ $\newline$ Choose $1$ answer: $\newline$ (A) $7983$ . $02$ $\newline$ (B) $13,301.03$ $\newline$ (C) $26,590.05$ $\newline$ (D) $39,891.08$

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Full solution

\sum_{k=0}^{19} 6(1.5)^{k} \approx

\newline

Choose

1

answer:

\newline

(A)

7983

02

\newline

(B)

13,301.03

\newline

(C)

26,590.05

\newline

(D)

39,891.08

Recognize as geometric series: Recognize the series as a geometric series where the first term $a = 6$ and the common ratio $r = 1.5$ . The sum of a finite geometric series can be calculated using the formula $S_n = a \frac{1 - r^n}{1 - r}$ , where $n$ is the number of terms.
Calculate number of terms: Calculate the number of terms $n$ . Since the series starts at $k=0$ and goes up to $k=19$ , there are $19 - 0 + 1 = 20$ terms.
Substitute values into formula: Substitute the values into the sum formula for a geometric series. Here, $a = 6$ , $r = 1.5$ , and $n = 20$ . So, $S_{20} = 6 \frac{1 - (1.5)^{20}}{1 - 1.5}$ .
Calculate sum: Calculate the sum $S_{20}$ . First, calculate $(1.5)^{20}$ and then substitute it into the formula.
Handle negative denominator: Since r > 1 , the denominator $1 - r$ is negative. Therefore, we need to be careful with the signs when calculating the sum. The correct formula application is $S_{20} = 6 \frac{1 - (1.5)^{20}}{1 - 1.5} = 6 \frac{1 - (1.5)^{20}}{-0.5}$ .
Compute ( $1$ . $5$ )^ $20$ : Compute $(1.5)^{20}$ using a calculator. $\newline$ $(1.5)^{20} \approx 33219.476736$
Substitute into formula and calculate: Substitute $(1.5)^{20}$ into the sum formula and calculate $S_{20}$ . $\newline$ $S_{20} = 6 \frac{1 - 33219.476736}{-0.5}$ $\newline$ $S_{20} = 6 \frac{-33218.476736}{-0.5}$ $\newline$ $S_{20} = 6 \cdot 66436.953472$ $\newline$ $S_{20} = 398621.720832$