sum_(k=0)^(19)-4(-2)^(k)~~ Choose 1 answer: (A) -699,052 (B) 1,398,100 (c) 2,097,152 (D) 4,194,308

Identify series type: Identify the type of series. The series is of the form -4(-2)^k, which is a geometric series because each term is obtained by multiplying the previous term by a common ratio.Determine first term: Determine the first term of the series. The first term is obtained when k=0, which gives us -4(-2)^0 = -4(1) = -4.Determine common ratio: Determine the common ratio of the series. The common ratio is -2, as each term is multiplied by -2 to get the next term.Use formula for sum: 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 S = a(1 - r^n) / (1 - r), provided that r is not equal to 1.Apply formula to series: Apply the formula to the given series. Here, a = -4, r = -2, and n = 20 (since we are summing from k=0 to k=19, which gives us 20 terms). S = -4(1 - (-2)^20) / (1 - (-2))Calculate sum: Calculate the sum. S = -4(1 - 2^20) / (1 + 2) S = -4(1 - 1048576) / 3 S = -4(-1048575) / 3 S = 4 * 1048575 / 3 S = 4194300 / 3Final calculation: Final calculation. S = 1398100

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

sum_(k=0)^(19)-4(-2)^(k)~~
Choose 1 answer:
(A)
-699,052
(B)
1,398,100
(c)
2,097,152
(D)
4,194,308

$\sum_{k=0}^{19}-4(-2)^{k} \approx$ $\newline$ Choose $1$ answer: $\newline$ (A) $-699,052$ $\newline$ (B) $1,398,100$ $\newline$ (C) $2,097,152$ $\newline$ (D) $4,194,308$

View step-by-step help

Full solution

\sum_{k=0}^{19}-4(-2)^{k} \approx

\newline

Choose

1

answer:

\newline

(A)

-699,052

\newline

(B)

1,398,100

\newline

(C)

2,097,152

\newline

(D)

4,194,308

Identify series type: Identify the type of series. $\newline$ The series is of the form $-4(-2)^k$ , which is a geometric series because each term is obtained by multiplying the previous term by a common ratio.
Determine first term: Determine the first term of the series. $\newline$ The first term is obtained when $k=0$ , which gives us $-4(-2)^0 = -4(1) = -4$ .
Determine common ratio: Determine the common ratio of the series. $\newline$ The common ratio is $-2$ , as each term is multiplied by $-2$ to get the next term.
Use formula for sum: 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 $S = \frac{a(1 - r^n)}{(1 - r)}$ , provided that $r \neq 1$ .
Apply formula to series: Apply the formula to the given series. $\newline$ Here, $a = -4$ , $r = -2$ , and $n = 20$ (since we are summing from $k=0$ to $k=19$ , which gives us $20$ terms). $\newline$ $S = -4(1 - (-2)^{20}) / (1 - (-2))$
Calculate sum: Calculate the sum. $\newline$ $S = -4(1 - 2^{20}) / (1 + 2)$ $\newline$ $S = -4(1 - 1048576) / 3$ $\newline$ $S = -4(-1048575) / 3$ $\newline$ $S = 4 \times 1048575 / 3$ $\newline$ $S = 4194300 / 3$
Final calculation: Final calculation. $\newline$ $S = 1398100$