Final answer:
The common ratio in the given infinite geometric sequence is 1/5. The sum of the infinite geometric series is 20. The values of n for which the sum of the first n terms is greater than 31 are n = 3 and any value greater than 3.
Step-by-step explanation:
An infinite geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this case, the common ratio can be found by dividing any term by the previous term. Let's use the second and first terms to find the common ratio: 3/16 = 1/5.
So, the common ratio is 1/5. To determine the values of n for which the sum of the first n terms is greater than 31, we can use the formula for the sum of an infinite geometric series: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
Let's substitute the given values into the formula:
S = 16 / (1 - 1/5) = 16 / (4/5) = 20.
This means that the sum of the infinite geometric series is 20. To find the values of n for which the sum is greater than 31, we need to find the partial sums of the series and determine when they exceed 31.
Let's write out the partial sums:
S1 = 16
S2 = 16 + 16/5 = 88/5
S3 = 16 + 16/5 + 16/25 = 448/25
We can see that S3 is greater than 31, so the possible values of n for which the sum of the first n terms is greater than 31 are n = 3 and any value greater than 3.