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From the table, you are asked to predict the population of India for the year 2040, with \( x = 90 \) (i.e., 90 years after 1950). To achieve this, we need to apply a consistent method based on the information provided.

### Steps to follow:
1. **Average Population Growth Rate Calculation**: 
   The table shows population increases in percentage over different decades. The method provided averages the growth rates across the decades: 

   \[
   \frac{0.26 + 0.25 + 0.26 + 0.26 + 0.22 + 0.18 + 0.16}{7} = 0.227 \text{ or } 22.7\%
   \]

2. **Annual Population Growth Rate**: 
   Since the percentage growth of 22.7% is per decade (10 years), the annual growth rate can be approximated as:

   \[
   \frac{0.227}{10} = 0.0227 \text{ or } 2.27\% \text{ per year}
   \]

3. **Current Population (2023)**: 
   The population in 2023 is \( 1,438,069,597 \).

4. **Time Span (2023 to 2040)**: 
   From 2023 to 2040, the time span is 17 years. 

5. **Exponential Growth Formula**: 
   To model population growth, we can use the exponential growth formula:

   \[
   P(t) = P_0 \times (1 + r)^t
   \]
   where:
   - \( P(t) \) is the future population,
   - \( P_0 \) is the current population (\( 1,438,069,597 \) in 2023),
   - \( r \) is the growth rate (2.27% or 0.0227),
   - \( t \) is the number of years (17 years from 2023 to 2040).

   Plugging in the values:

   \[
   P(17) = 1,438,069,597 \times (1 + 0.0227)^{17}
   \]

6. **Calculation**:
   First, calculate the growth factor:

   \[
   (1 + 0.0227)^{17} \approx 1.478
   \]

   Now, multiply this by the 2023 population:

   \[
   P(17) \approx 1,438,069,597 \times 1.478 \approx 2,124,544,688
   \]

So, the predicted population of India in 2040 is approximately **2.12 billion**.

---

Let me know if you would like more details on any step!

### Questions to Expand on:
1. How does the annual growth rate impact predictions over longer time periods?
2. What are alternative models for predicting population growth, and how do they compare to exponential models?
3. Can population growth slow down, and what factors might lead to this in real-world scenarios?
4. How would changes in birth rate, death rate, and immigration affect future predictions?
5. What are the limitations of using historical data for predicting future populations?

### Tip:
Always check if the growth rate is consistent over time—different decades may experience varying trends in growth!

Predict the population of India in the year 2040 based on historical population data using an exponential growth model.

Learn how to predict India's population for the year 2040 using an exponential growth model. This guide covers population growth trends from 1950 to 2023, calculating the average growth rate, and applying the exponential growth formula. Suitable for high school students studying exponential growth and population modeling.

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