2. Find the sum of the following series. Round to the nearest hundredth if necessary.
$6 + 12 + 24 + \dots + 1536$
Sum of a finite geometric series:
$S_{n} = \frac{a_{1} - a_{1} r^{n}}{1 - r}$

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The sum of the series is $3066$

Steps

Step 1 ：Given the geometric series $6 + 12 + 24 + \dots + 1536$

Step 2 ：We can see that the common ratio (r) is 2 (each term is twice the previous term) and the first term (a1) is 6.

Step 3 ：The number of terms (n) can be found by dividing the last term by the first term and taking the base-2 logarithm, then adding 1. So, $n = \log_{2} (\frac{1536}{6}) + 1 = 9.0$

Step 4 ：We can use the formula for the sum of a geometric series to find the sum: $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}$

Step 5 ：Substitute the values into the formula: $S_{n} = \frac{6 (1 - 2^{9})}{1 - 2} = 3066.0$

Step 6 ：Final Answer: The sum of the series is $3066$

2. Find the sum of the following series. Round to the nearest hundredth if necessary.6+12+24+…+1536Sum of a finite geometric series:Sn=a1−a1rn1−r

Answer

Popular Problems

2. Find the sum of the following series. Round to the nearest hundredth if necessary.
$6 + 12 + 24 + \dots + 1536$
Sum of a finite geometric series:
$S_{n} = \frac{a_{1} - a_{1} r^{n}}{1 - r}$