Find the sum of the first 7 terms of the following sequence. Round to the nearest hundredth if necessary. 92,quad78.2,quad66.47,dots Sum of a finite geometric series: S_(n)=(a_(1)-a_(1)r^(n))/(1-r) Answer:

Identify common ratio: First, we need to identify the common ratio (r) of the geometric sequence. To do this, we divide the second term by the first term. r = 78.2 / 92 r ≈ 0.85 Calculate sum of first 7 terms: Now that we have the common ratio, we can use the formula for the sum of the first n terms of a geometric series to find the sum of the first 7 terms. S_7 = (a_1 - a_1 * r^7) / (1 - r)Substitute values into formula: We substitute the values we know into the formula. The first term a_1 is 92, and r is approximately 0.85. S_7 = (92 - 92 * 0.85^7) / (1 - 0.85)Calculate r^7: Now we calculate the value of r^7. 0.85^7 ≈ 0.320577Perform calculations: We substitute this value back into the formula for S_7. S_7 = (92 - 92 * 0.320577) / (1 - 0.85)Calculate denominator: We perform the calculations inside the parentheses. 92 * 0.320577 ≈ 29.49308 92 - 29.49308 ≈ 62.50692Find S_7: Now we calculate the denominator of the formula. 1 - 0.85 = 0.15Round to nearest hundredth: Finally, we divide the numerator by the denominator to find S_7. S_7 = 62.50692 / 0.15 S_7 ≈ 416.7128We round the sum to the nearest hundredth as instructed. S_7 ≈ 416.71

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Q. Find the sum of the first

7

terms of the following sequence. Round to the nearest hundredth if necessary.

\newline

92, \quad 78.2, \quad 66.47, \ldots

\newline

Sum of a finite geometric series:

\newline

S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}

\newline

Answer:

Identify common ratio: First, we need to identify the common ratio $r$ of the geometric sequence. To do this, we divide the second term by the first term. $r = \frac{78.2}{92}$ $r \approx 0.85$
Calculate sum of first $7$ terms: Now that we have the common ratio, we can use the formula for the sum of the first $n$ terms of a geometric series to find the sum of the first $7$ terms. $\newline$ $S_7 = \frac{(a_1 - a_1 \cdot r^7)}{(1 - r)}$
Substitute values into formula: We substitute the values we know into the formula. The first term $a_1$ is $92$ , and $r$ is approximately $0.85$ . $S_7 = \frac{92 - 92 \times 0.85^7}{1 - 0.85}$
Calculate $r^7$ : Now we calculate the value of $r^7$ . $\newline$ $0.85^7 \approx 0.320577$
Perform calculations: We substitute this value back into the formula for $S_7$ . $\newline$ $S_7 = \frac{(92 - 92 \times 0.320577)}{(1 - 0.85)}$
Calculate denominator: We perform the calculations inside the parentheses. $\newline$ $92 \times 0.320577 \approx 29.49308$ $\newline$ $92 - 29.49308 \approx 62.50692$
Find $S_7$ : Now we calculate the denominator of the formula. $\newline$ $1 - 0.85 = 0.15$
Round to nearest hundredth: Finally, we divide the numerator by the denominator to find $S_7$ . $\newline$ $S_7 = \frac{62.50692}{0.15}$ $\newline$ $S_7 \approx 416.7128$
Round to nearest hundredth: Finally, we divide the numerator by the denominator to find $S_7$ . $\newline$ $S_7 = \frac{62.50692}{0.15}$ $\newline$ $S_7 \approx 416.7128$ We round the sum to the nearest hundredth as instructed. $\newline$ $S_7 \approx 416.71$