Questions: Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first five terms of the geometric sequence: 3/3, 3/6, 3/12, .... -1/12 31/16 93 1/12

Solution Steps

To find the sum of the first five terms of a geometric sequence, we need to identify the first term (a) and the common ratio (r). Then, we use the formula for the sum of the first n terms of a geometric sequence: \( S_n = a \frac{1-r^n}{1-r} \).

1. Identify the first term (a) and the common ratio (r).
2. Use the formula \( S_n = a \frac{1-r^n}{1-r} \) to find the sum of the first five terms.

The first term of the geometric sequence is given by: \[ a = \frac{3}{3} = 1.0 \] The common ratio can be calculated as: \[ r = \frac{\frac{3}{6}}{\frac{3}{3}} = \frac{1}{2} = 0.5 \]

We apply the formula for the sum of the first \( n \) terms of a geometric sequence: \[ S_n = a \frac{1 - r^n}{1 - r} \] Substituting the values \( a = 1.0 \), \( r = 0.5 \), and \( n = 5 \): \[ S_5 = 1.0 \cdot \frac{1 - (0.5)^5}{1 - 0.5} \]

Calculating \( (0.5)^5 \): \[ (0.5)^5 = 0.03125 \] Now substituting this back into the sum formula: \[ S_5 = 1.0 \cdot \frac{1 - 0.03125}{0.5} = 1.0 \cdot \frac{0.96875}{0.5} = 1.0 \cdot 1.9375 = 1.9375 \]

Final Answer