To complete the series 5, 13, 45, 189, 957, we need to identify the pattern or rule governing the sequence.
Let's look at the differences between consecutive terms:
13 - 5 = 8
45 - 13 = 32
189 - 45 = 144
957 - 189 = 768
The differences between the consecutive terms are not consistent. Therefore, the series does not follow a simple arithmetic or geometric progression.
However, if we look at the differences between the differences, we can see a pattern:
32 - 8 = 24
144 - 32 = 112
768 - 144 = 624
The differences between the differences are increasing by a factor of 3. Therefore, we can assume that the next difference should be 624 * 3 = 1872.
Now, let's calculate the next term in the series:
957 + 1872 = 2829
Among the given options, none of them match the calculated value of 2829. Therefore, none of the given options is the correct answer.
It's possible that there is a mistake in the given series, or the series follows a pattern that is not immediately obvious.