According to a census report, the population growth rate of Faridabad is going to be an increasing A.P. with first year's rate as 5% and common difference as 5%, but simultaneously the migration rate is an increasing G.P. with first term as 1% and common ratio of 2. If population on 31 December 2010 is 1 million, then find in which year will Faridabad witness its first fall in population?

Options

A. 2016
B. 2014
C. 2013
D. 2019

Answer

To solve the problem, we need to analyze the population growth rate and the migration rate year by year until we find the year when the population starts to decline.

### Step-by-Step Solution:

1. **Identify the Initial Population and Rates**:
- Initial population on December 31, 2010 = 1,000,000.
- Population growth rate (increasing A.P.):
- First year = 5%
- Common difference = 5%
- Migration rate (increasing G.P.):
- First term = 1%
- Common ratio = 2

2. **Calculate the Population Growth Rate for Each Year**:
- Year 1 (2011): Growth = 5%
- Year 2 (2012): Growth = 5% + 5% = 10%
- Year 3 (2013): Growth = 10% + 5% = 15%
- Year 4 (2014): Growth = 15% + 5% = 20%
- Year 5 (2015): Growth = 20% + 5% = 25%
- Year 6 (2016): Growth = 25% + 5% = 30%

3. **Calculate the Migration Rate for Each Year**:
- Year 1 (2011): Migration = 1%
- Year 2 (2012): Migration = 1% * 2 = 2%
- Year 3 (2013): Migration = 2% * 2 = 4%
- Year 4 (2014): Migration = 4% * 2 = 8%
- Year 5 (2015): Migration = 8% * 2 = 16%
- Year 6 (2016): Migration = 16% * 2 = 32%

4. **Calculate the Effective Population Change for Each Year**:
- Year 1 (2011):
- Population increase = 5% of 1,000,000 = 50,000
- Migration = 1% of 1,000,000 = 10,000
- Net population = 1,000,000 + 50,000 - 10,000 = 1,040,000
- Year 2 (2012):
- Population increase = 10% of 1,040,000 = 104,000
- Migration = 2% of 1,040,000 = 20,800
- Net population = 1,040,000 + 104,000 - 20,800 = 1,123,200
- Year 3 (2013):
- Population increase = 15% of 1,123,200 = 168,480
- Migration = 4% of 1,123,200 = 44,928
- Net population = 1,123,200 + 168,480 - 44,928 = 1,246,752
- Year 4 (2014):
- Population increase = 20% of 1,246,752 = 249,350.4
- Migration = 8% of 1,246,752 = 99,740.16
- Net population = 1,246,752 + 249,350.4 - 99,740.16 = 1,396,362.24
- Year 5 (2015):
- Population increase = 25% of 1,396,362.24 = 349,090.56
- Migration = 16% of 1,396,362.24 = 223,417.96
- Net population = 1,396,362.24 + 349,090.56 - 223,417.96 = 1,521,034.84
- Year 6 (2016):
- Population increase = 30% of 1,521,034.84 = 456,310.45
- Migration = 32% of 1,521,034.84 = 486,712.74
- Net population = 1,521,034.84 + 456,310.45 - 486,712.74 = 1,490,632.55

5. **Determine the Year of First Population Decline**:
- In 2016, the population increase (30%) is less than the migration (32%). Therefore, the first fall in population will occur in the year 2016.

### Final Answer:
Faridabad will witness its first fall in population in the year **2016**.