Find the sum of the first $45$ terms in this geometric series: $\newline$ $-0.5+1.5-4.5 \ldots$ $\newline$ Choose $1$ answer: $\newline$ (A) $-7.39 \cdot 10^{20}$ $\newline$ (B) $-4.92 \cdot 10^{20}$ $\newline$ (C) $-3.69 \cdot 10^{20}$ $\newline$ (D) $1.23 \cdot 10^{20}$

View step-by-step help

Home

Math Problems

Grade 8

Multiply numbers written in scientific notation

Full solution

Q. Find the sum of the first

45

terms in this geometric series:

\newline

-0.5+1.5-4.5 \ldots

\newline

Choose

1

answer:

\newline

(A)

-7.39 \cdot 10^{20}

\newline

(B)

-4.92 \cdot 10^{20}

\newline

(C)

-3.69 \cdot 10^{20}

\newline

(D)

1.23 \cdot 10^{20}

Identify terms and ratio: Identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a = -0.5$ , and each term is multiplied by $-3$ to get the next term (since $-0.5 \times -3 = 1.5$ , $1.5 \times -3 = -4.5$ , and so on), so the common ratio $r = -3$ .
Use sum formula: Use the formula for the sum of the first $n$ terms of a geometric series: $S_n = \frac{a(1 - r^n)}{(1 - r)}$ , where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ Here, $n = 45$ , $a = -0.5$ , and $r = -3$ .
Calculate denominator: Substitute the values into the formula and calculate the sum. $S_{45} = -0.5(1 - (-3)^{45}) / (1 - (-3))$
Calculate exponent: Calculate the denominator of the fraction: $1 - (-3) = 1 + 3 = 4$ .
Estimate numerator: Calculate $(-3)^{45}$ . Since $45$ is an odd number, the result will be negative, and the magnitude will be $3^{45}$ .
Substitute values: The magnitude of $3^{45}$ is a very large number, and it's not practical to calculate it exactly without a calculator. However, we can estimate that it will be a number much larger than $10^{20}$ , and since it's raised to an odd power, the result will be negative.
Consider magnitude: Now, calculate the numerator of the fraction: $1 - (-3)^{45}$ . Since $(-3)^{45}$ is negative and its magnitude is much larger than $1$ , the result will be approximately equal to $-(-3)^{45}$ .
Eliminate options: Substitute the calculated values into the sum formula: $\newline$ $S_{45} \approx -0.5 \times (-(-3)^{45}) / 4$ $\newline$ $S_{45} \approx 0.5 \times 3^{45} / 4$
Choose closest option: Since $3^{45}$ is much larger than $4$ , dividing by $4$ will not significantly affect the magnitude of the number. Therefore, the sum will be approximately $0.5 \times 3^{45}.$
Choose closest option: Since $3^{45}$ is much larger than $4$ , dividing by $4$ will not significantly affect the magnitude of the number. Therefore, the sum will be approximately $0.5 \times 3^{45}$ .Given the choices, we are looking for a negative number with a magnitude in the order of $10^{20}$ . The only negative options are (A), (B), and (C).
Choose closest option: Since $3^{45}$ is much larger than $4$ , dividing by $4$ will not significantly affect the magnitude of the number. Therefore, the sum will be approximately $0.5 \times 3^{45}$ .Given the choices, we are looking for a negative number with a magnitude in the order of $10^{20}$ . The only negative options are (A), (B), and (C).Since the sum is approximately $0.5 \times 3^{45}$ and we know $3^{45}$ is a very large number, we can eliminate the smallest magnitude among the negative options, which is (C) $-3.69\times10^{20}$ .
Choose closest option: Since $3^{45}$ is much larger than $4$ , dividing by $4$ will not significantly affect the magnitude of the number. Therefore, the sum will be approximately $0.5 \times 3^{45}$ .Given the choices, we are looking for a negative number with a magnitude in the order of $10^{20}$ . The only negative options are (A), (B), and (C).Since the sum is approximately $0.5 \times 3^{45}$ and we know $3^{45}$ is a very large number, we can eliminate the smallest magnitude among the negative options, which is (C) $-3.69\times10^{20}$ .Between (A) and (B), we choose the one that is closest to half of $3^{45}$ . Without an exact calculation, we cannot determine whether (A) $-7.39\times10^{20}$ or (B) $4$ $0$ is the correct answer. However, since we are dividing by $4$ in the last step, it is more likely that the sum is closer to (B) $4$ $0$ than to (A) $-7.39\times10^{20}$ .