Find the sum of the first $6$ terms of the following sequence. Round to the nearest hundredth if necessary. $\newline$ $72, \quad 68.4, \quad 64.98, \ldots$ $\newline$ Sum of a finite geometric series: $\newline$ $S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}$ $\newline$ Answer:

Full solution

Q. Find the sum of the first

6

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

\newline

72, \quad 68.4, \quad 64.98, \ldots

\newline

Sum of a finite geometric series:

\newline

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

\newline

Answer:

Find Common Ratio: First, we need to identify the common ratio $r$ of the geometric sequence. We can find the common ratio by dividing the second term by the first term. $\newline$ Calculation: $r = \frac{68.4}{72}$
Calculate Common Ratio: Now, let's perform the calculation to find the common ratio. $\newline$ Calculation: $r = \frac{68.4}{72} = 0.95$
Use Geometric Series Formula: Next, we will use the formula for the sum of the first $n$ terms of a geometric series, which is $S_n = \frac{a_1 - a_1 \cdot r^n}{1 - r}$ , where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ We have $a_1 = 72$ , $r = 0.95$ , and $n = 6$ .
Substitute Values and Calculate: Now, we will substitute the values into the formula and calculate the sum of the first $6$ terms. $\newline$ Calculation: $S_6 = \frac{72 - 72 \times 0.95^6}{1 - 0.95}$
Perform Calculation for Sum: Let's perform the calculation for the sum. $\newline$ Calculation: $S_6 = \frac{(72 - 72 \times 0.95^6)}{(1 - 0.95)} = \frac{(72 - 72 \times 0.7350918906249995)}{0.05}$
Calculate Exact Sum: Now, we will calculate the exact value of the sum. $\newline$ Calculation: $S_6 = (72 - 52.92660998379997) / 0.05 = 19.07339001620003 / 0.05 = 381.4678003240006$
Round to Nearest Hundredth: Finally, we will round the sum to the nearest hundredth as instructed. $\newline$ Calculation: $S_6 \approx 381.47$