Final answer:
To find the sum of an infinite geometric series, you can use the formula S = a / (1 - r), where a is the first term and r is the common ratio. Options a, b, c, and d all have an infinite geometric series and can be solved using this formula.
Step-by-step explanation:
The formula for finding the sum of an infinite geometric series is S = a / (1 - r), where a is the first term and r is the common ratio.
Looking at the given options:
- a. The first term is 2 and the common ratio is 2, so the sum can be found using the formula.
- b. The first term is 3 and the common ratio is 2, so the sum can be found using the formula.
- c. The first term is 5 and the common ratio is 2, so the sum can be found using the formula.
- d. The first term is 7 and the common ratio is 2, so the sum can be found using the formula.
The infinite series a. 2 + 4 + 8 + 16 + ..., b. 3 + 6 + 12 + 24 + ..., c. 5 + 10 + 20 + 40 + ..., and d. 7 + 14 + 28 + 56 + ... are all geometric series with a common ratio that is greater than 1. When dealing with an infinite geometric series, you can find the sum using a formula only if the common ratio (r) is between -1 and 1, exclusive.
As all the series presented here have a common ratio (r) greater than 1, none of these series converge, and hence, you cannot find a sum for these series using the standard geometric series sum formula.Therefore, the infinite series for which you would use a formula to find the sum are options a, b, c, and d.