Questions: 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.) 7.0001 c.) 7.6923 d) 32.8719

Solution Steps

To find the sum of the first 12 terms of a geometric sequence, we use the formula for the sum of the first \( n \) terms of a geometric sequence:

\[ S_n = a \frac{1 - r^n}{1 - r} \]

where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms. Here, \( a = 10 \), \( r = 0.3 \), and \( n = 12 \). We will substitute these values into the formula to calculate the sum.

To find the sum of the first \( n \) terms of a geometric sequence, we use the formula:

\[ S_n = a \frac{1 - r^n}{1 - r} \]

where:

• \( a \) is the first term,
• \( r \) is the common ratio,
• \( n \) is the number of terms.

Given:

• \( a = 10 \)
• \( r = 0.3 \)
• \( n = 12 \)

Substitute these values into the formula:

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

Calculate the expression:

\[ S_{12} = 10 \frac{1 - 0.000000531441}{0.7} \]

\[ S_{12} = 10 \frac{0.999999468559}{0.7} \]

\[ S_{12} \approx 14.2857 \]

Final Answer