Final answer:
The pattern in the sequence √3, √12, √27, √48, ... corresponds to √(3n^2) where n is a consecutive natural number starting from 1. The next term in the series is √75.
Step-by-step explanation:
To complete the pattern √3, √12, √27, √48, ......, we need to find the sequence that governs this series. Observing the square roots, we see that the numbers under the square root are 3, 12, 27, and 48 which are 3 times the square numbers 1, 4, 9, and 16 respectively. Thus, the pattern is given by √(3n^2) where n is an integer sequence starting from 1. The next term would be for n = 5. Upon squaring 5, we get 25, and then multiplying by 3 we get 75. Therefore, the next term in the pattern is √75.