The sum of the first 12 terms of a geometric sequence with
10 as the first term and a common ratio of 0.3 is
a.) 14.2857
b.) 7.6923
c.) 32.8719
d.) 7.0001

Question

The sum of the first 12 terms of a geometric sequence with 10 as the first term and a common ratio of 0.3 is a.) 14.2857 b.) 7.6923 c.) 32.8719 d.) 7.0001

UpStudy AI · Accepted Answer

To find the sum of the first 12 terms of a geometric sequence, we can use the formula for the sum 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
- $$ n $$ is the number of terms

Given:
- First term ($$ a $$) = 10
- Common ratio ($$ r $$) = 0.3
- Number of terms ($$ n $$) = 12

Substitute the given values into the formula:

$$ S_{12} = \frac{10(1 - 0.3^{12})}{1 - 0.3} $$

Now, we can calculate the sum of the first 12 terms of the geometric sequence.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{10\left(1-0.3^{12}\right)}{\left(1-0.3\right)}$$
- step1: Remove the parentheses:
  $$\frac{10\left(1-0.3^{12}\right)}{1-0.3}$$
- step2: Convert the expressions:
  $$\frac{10\left(1-\left(\frac{3}{10}\right)^{12}\right)}{1-0.3}$$
- step3: Subtract the numbers:
  $$\frac{10\times \frac{10^{12}-3^{12}}{10^{12}}}{1-0.3}$$
- step4: Multiply the numbers:
  $$\frac{\frac{10^{12}-3^{12}}{10^{11}}}{1-0.3}$$
- step5: Subtract the numbers:
  $$\frac{\frac{10^{12}-3^{12}}{10^{11}}}{0.7}$$
- step6: Convert the expressions:
  $$\frac{\frac{10^{12}-3^{12}}{10^{11}}}{\frac{7}{10}}$$
- step7: Multiply by the reciprocal:
  $$\frac{10^{12}-3^{12}}{10^{11}}\times \frac{10}{7}$$
- step8: Reduce the numbers:
  $$\frac{10^{12}-3^{12}}{10^{10}}\times \frac{1}{7}$$
- step9: Multiply the fractions:
  $$\frac{10^{12}-3^{12}}{10^{10}\times 7}$$
- step10: Multiply:
  $$\frac{10^{12}-3^{12}}{7\times 10^{10}}$$
The sum of the first 12 terms of the geometric sequence is approximately 14.2857.

Therefore, the correct answer is a.) 14.2857.
The sum of the first 12 terms is approximately 14.2857.

The sum of the first 12 terms of a geometric sequence with
10 as the first term and a common ratio of 0.3 is
a.) 14.2857
b.) 7.6923
c.) 32.8719
d.) 7.0001

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