Correct Option (D)
If 1/3rd of the population migrates each year, the proportion of the population remaining is 1 - 1/3 = 2/3 of the previous year's population. This process is repeated annually for six years, with no additional population growth.
The population remaining after the first year will be (2/3) of the initial population.
After the second year, the remaining population will be (2/3) × (2/3) = (2/3)² of the initial population.
Following this geometric progression, the population remaining after the sixth year will be (2/3)⁶ of the initial population.
Calculating the value: (2/3)⁶ = 2⁶ / 3⁶ = 64 / 729.
Therefore, 64/729th part of the initial population remains after the sixth year.
Incorrect Options:
Options A, B, and C are incorrect as they represent different powers or incorrect calculations of the remaining population fraction over the specified period.
- Option A (16/243rd part): This value does not correspond to the correct calculation of (2/3)⁶. For instance, 16/243 is equivalent to 2⁴/3⁵.
- Option B (32/243rd part): This value corresponds to (2/3)⁵, which would be the population remaining after the fifth year, not the sixth.
- Option C (32/729th part): This value corresponds to 2⁵/3⁶, indicating a miscalculation in either the numerator's or the denominator's power.