To calculate the sum of an infinite geometric series, we first need to determine whether the series converges. An infinite geometric series converges if the absolute value of the common ratio, r, is less than 1 (|r| < 1). If it converges, the sum can be calculated using the formula:
S = a / (1 – r)
where S is the sum of the series, a is the first term, and r is the common ratio.
Let’s analyze the given series: 3, 32, 34, 38, 316. First, we need to assess whether this is a geometric series by examining the ratio of successive terms:
- Ratio of second term to first term: r1 = 32 / 3 ≈ 10.67
- Ratio of third term to second term: r2 = 34 / 32 ≈ 1.0625
- Ratio of fourth term to third term: r3 = 38 / 34 ≈ 1.1176
- Ratio of fifth term to fourth term: r4 = 316 / 38 ≈ 8.3158
Since the ratios are not constant, this series is not geometric, indicating that we cannot apply the formula for the sum of an infinite geometric series. Therefore, we cannot determine the sum as if it were an infinite geometric series.
In conclusion, since the series does not fit the definition of a geometric series, we cannot calculate its sum as an infinite geometric series using the standard sum formula. The values provided do not create a consistent ratio, which is essential for applying the infinite series sum formula.