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

Identify series type: Identify the type of series. The given series is a geometric series because each term is obtained by multiplying the previous term by a common ratio, which is 1.5 in this case.Use sum formula: Use the formula for the sum of a finite geometric series. The sum S of a geometric series with first term a, common ratio r, and n terms is given by the formula: S = a(1 - r^n) / (1 - r), where r ≠ 1. In this problem, a = 6, r = 1.5, and n = 20 (since we start counting from k=0).Calculate sum: Plug the values into the formula and calculate the sum. S = 6(1 - 1.5^20) / (1 - 1.5) First, calculate 1.5^20.Calculate 1.5^20: Calculate 1.5^20. Using a calculator, we find that 1.5^20 ≈ 35429.41.Substitute into sum formula: Substitute 1.5^20 into the sum formula. S = 6(1 - 35429.41) / (1 - 1.5) S = 6(-35428.41) / (-0.5)Check for errors: Calculate the sum. S = 6 * -35428.41 / -0.5 S = -212570.46 / -0.5 S = 425140.92Re-evaluate calculation: Check for any possible errors in calculation. Re-evaluate the calculations to ensure there are no mistakes. Everything seems correct.Re-evaluate the calculation for any possible errors. Upon re-evaluation, it is noticed that the sum S was incorrectly calculated as 425140.92, which is not an option in the multiple-choice answers. The correct calculation should be: S = 6 * -35428.41 / -0.5 S = -212570.46 / -0.5 S = 425140.92 / 2 S = 212570.46 This value still does not match any of the choices provided. It seems there has been a mistake in the calculation process.

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Let’s check out your problem:

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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Math Problems

Algebra 1

Compare linear and exponential growth

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

Identify series type: Identify the type of series. $\newline$ The given series is a geometric series because each term is obtained by multiplying the previous term by a common ratio, which is $1.5$ in this case.
Use sum formula: Use the formula for the sum of a finite geometric series. $\newline$ The sum $S$ of a geometric series with first term $a$ , common ratio $r$ , and $n$ terms is given by the formula: $\newline$ $S = \frac{a(1 - r^n)}{(1 - r)},$ where $r \neq 1$ . $\newline$ In this problem, $a = 6$ , $r = 1.5$ , and $n = 20$ (since we start counting from $k=0$ ).
Calculate sum: Plug the values into the formula and calculate the sum. $\newline$ $S = 6(1 - 1.5^{20}) / (1 - 1.5)$ $\newline$ First, calculate $1.5^{20}$ .
Calculate $1.5^{20}$ : Calculate $1.5^{20}$ . Using a calculator, we find that $1.5^{20} \approx 35429.41$ .
Substitute into sum formula: Substitute $1.5^{20}$ into the sum formula. $\newline$ $S = 6(1 - 35429.41) / (1 - 1.5)$ $\newline$ $S = 6(-35428.41) / (-0.5)$
Check for errors: Calculate the sum. $\newline$ $S = 6 \times -35428.41 / -0.5$ $\newline$ $S = -212570.46 / -0.5$ $\newline$ $S = 425140.92$
Re-evaluate calculation: Check for any possible errors in calculation. $\newline$ Re-evaluate the calculations to ensure there are no mistakes. Everything seems correct.
Re-evaluate calculation: Check for any possible errors in calculation. $\newline$ Re-evaluate the calculations to ensure there are no mistakes. Everything seems correct.Re-evaluate the calculation for any possible errors. $\newline$ Upon re-evaluation, it is noticed that the sum $S$ was incorrectly calculated as $425140.92$ , which is not an option in the multiple-choice answers. The correct calculation should be: $\newline$ $S = 6 \times -35428.41 / -0.5$ $\newline$ $S = -212570.46 / -0.5$ $\newline$ $S = 425140.92 / 2$ $\newline$ $S = 212570.46$ $\newline$ This value still does not match any of the choices provided. It seems there has been a mistake in the calculation process.