According to the 2001 census, the population growth rate of Lucknow is going to be an increasing. AP with first year’s rate as 5% and common difference as 5%, but simultaneously the migration, rate is an increasing GP with first term as 1% and common ratio as 2. If population on 31 December 2000 is 1 million, then find in which year will Lucknow witness its first fall in population?

Options

A. 2005
B. 2006
C. 2007
D. 2008

Answer

To find out in which year Lucknow will witness its first fall in population, we need to analyze both the population growth rate and the migration rate.

### Step-by-Step Solution:

1. **Understanding the Population Growth Rate**:
- The population growth rate is given as an Arithmetic Progression (AP) with:
- First term (a) = 5%
- Common difference (d) = 5%
- The growth rates for the first few years will be:
- Year 1: 5%
- Year 2: 10%
- Year 3: 15%
- Year 4: 20%
- Year 5: 25%
- Year 6: 30%

2. **Understanding the Migration Rate**:
- The migration rate is given as a Geometric Progression (GP) with:
- First term (a) = 1%
- Common ratio (r) = 2
- The migration rates for the first few years will be:
- Year 1: 1%
- Year 2: 2%
- Year 3: 4%
- Year 4: 8%
- Year 5: 16%
- Year 6: 32%

3. **Calculating the Net Population Change**:
- The initial population on 31 December 2000 is 1 million.
- We will calculate the net population change for each year by subtracting the migration rate from the population growth rate.

4. **Calculating Year by Year**:
- **Year 2001**:
- Population growth = 5% of 1,000,000 = 50,000
- Migration = 1% of 1,000,000 = 10,000
- Net change = 50,000 - 10,000 = 40,000
- New population = 1,000,000 + 40,000 = 1,040,000

- **Year 2002**:
- Population growth = 10% of 1,040,000 = 104,000
- Migration = 2% of 1,040,000 = 20,800
- Net change = 104,000 - 20,800 = 83,200
- New population = 1,040,000 + 83,200 = 1,123,200

- **Year 2003**:
- Population growth = 15% of 1,123,200 = 168,480
- Migration = 4% of 1,123,200 = 44,928
- Net change = 168,480 - 44,928 = 123,552
- New population = 1,123,200 + 123,552 = 1,246,752

- **Year 2004**:
- Population growth = 20% of 1,246,752 = 249,350.40
- Migration = 8% of 1,246,752 = 99,740.16
- Net change = 249,350.40 - 99,740.16 = 149,610.24
- New population = 1,246,752 + 149,610.24 = 1,396,362.24

- **Year 2005**:
- Population growth = 25% of 1,396,362.24 = 349,090.56
- Migration = 16% of 1,396,362.24 = 223,417.96
- Net change = 349,090.56 - 223,417.96 = 125,672.60
- New population = 1,396,362.24 + 125,672.60 = 1,522,034.84

- **Year 2006**:
- Population growth = 30% of 1,522,034.84 = 456,610.45
- Migration = 32% of 1,522,034.84 = 486,726.75
- Net change = 456,610.45 - 486,726.75 = -30,116.30
- New population = 1,522,034.84 - 30,116.30 = 1,491,918.54

5. **Conclusion**:
- In the year 2006, the net population change becomes negative for the first time, indicating that Lucknow will witness its first fall in population.

### Final Answer:
Lucknow will witness its first fall in population in the year **2006**.