Find the sum of the geometric series $\newline$ $1-3+3^{2}-3^{3}+\ldots-3^{29}$ $\newline$ Choose $1$ answer: $\newline$ (A) $-1.03 \cdot 10^{14}$ $\newline$ (B) $-6.86 \cdot 10^{13}$ $\newline$ (C) $-5.15 \cdot 10^{13}$ $\newline$ (D) $1.72 \cdot 10^{13}$

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Grade 8

Add and subtract numbers written in scientific notation

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Q. Find the sum of the geometric series

\newline

1-3+3^{2}-3^{3}+\ldots-3^{29}

\newline

Choose

1

answer:

\newline

(A)

-1.03 \cdot 10^{14}

\newline

(B)

-6.86 \cdot 10^{13}

\newline

(C)

-5.15 \cdot 10^{13}

\newline

(D)

1.72 \cdot 10^{13}

Identify Geometric Series: The given series is a geometric series with the first term $a = 1$ and the common ratio $r = -3$ . The sum of a finite geometric series can be calculated using the formula $S_n = \frac{a(1 - r^n)}{1 - r}$ , where $n$ is the number of terms.
Determine Number of Terms: First, we need to determine the number of terms in the series. Since the series starts at $3^0$ (which is $1$ ) and goes up to $3^{29}$ , and the powers of $3$ increase by $1$ each time, there are $30$ terms in total.
Apply Sum Formula: Now we can apply the formula for the sum of a geometric series: $S_n = a(1 - r^n) / (1 - r)$ . Here, $a = 1$ , $r = -3$ , and $n = 30$ .
Calculate Numerator: Plugging the values into the formula, we get $S_{30} = \frac{1(1 - (-3)^{30})}{1 - (-3)}$ .
Calculate Denominator: Calculating the numerator: $1 - (-3)^{30} = 1 - 3^{30}$ . Since $3^{30}$ is a very large number, we can leave it in the exponential form for now.
Substitute Values: Calculating the denominator: $1 - (-3) = 1 + 3 = 4$ .
Calculate Exponential Value: Now we have $S_{30} = \frac{1 - 3^{30}}{4}$ . To find the exact value of $3^{30}$ , we can use a calculator or a computer.
Perform Subtraction: Using a calculator, we find that $3^{30}$ is approximately $2.05 \times 10^{14}$ .
Divide by $4$ : Substituting this value into our sum expression, we get $S_{30} = \frac{(1 - 2.05 \times 10^{14})}{4}$ .
Express in Scientific Notation: Now we perform the subtraction in the numerator: $1 - 2.05 \times 10^{14} = -2.05 \times 10^{14} + 1$ . Since $1$ is negligible compared to $2.05 \times 10^{14}$ , we can approximate this as $-2.05 \times 10^{14}$ .
Express in Scientific Notation: Now we perform the subtraction in the numerator: $1 - 2.05 \times 10^{14} = -2.05 \times 10^{14} + 1$ . Since $1$ is negligible compared to $2.05 \times 10^{14}$ , we can approximate this as $-2.05 \times 10^{14}$ .Finally, we divide by $4$ : $S_{30} = -2.05 \times 10^{14} / 4 = -0.5125 \times 10^{14}$ .
Express in Scientific Notation: Now we perform the subtraction in the numerator: $1 - 2.05 \times 10^{14} = -2.05 \times 10^{14} + 1$ . Since $1$ is negligible compared to $2.05 \times 10^{14}$ , we can approximate this as $-2.05 \times 10^{14}$ .Finally, we divide by $4$ : $S_{30} = -2.05 \times 10^{14} / 4 = -0.5125 \times 10^{14}$ .We can write $-0.5125 \times 10^{14}$ in scientific notation as $-5.125 \times 10^{13}$ .