Questions: Find the sum of the infinite geometric sequence 450, 180, 72, 28.8, 11.52 ... Sum =

Solution Steps

To find the sum of an infinite geometric sequence, we need to identify the first term \( a \) and the common ratio \( r \). The sum \( S \) of an infinite geometric sequence can be calculated using the formula \( S = \frac{a}{1 - r} \), provided that \( |r| < 1 \).

1. Identify the first term \( a \).
2. Calculate the common ratio \( r \) by dividing the second term by the first term.
3. Use the formula \( S = \frac{a}{1 - r} \) to find the sum.

The first term of the geometric sequence is given as: \[ a = 450 \]

The common ratio \( r \) is calculated by dividing the second term by the first term: \[ r = \frac{180}{450} = 0.4 \]

For the sum of an infinite geometric series to exist, the absolute value of the common ratio must be less than 1: \[ |r| = |0.4| < 1 \] Since this condition is satisfied, we can proceed to calculate the sum.

The sum \( S \) of an infinite geometric series is given by the formula: \[ S = \frac{a}{1 - r} \] Substituting the values of \( a \) and \( r \): \[ S = \frac{450}{1 - 0.4} = \frac{450}{0.6} = 750 \]

Final Answer