The last number in the expression
`4 + 9^2 + 4^3 + 9^4 + 4^5 + 9^6 + ..... + 4^99 + 9^100 `

Options

A. 0
B. 3
C. 5
D. None of these

Answer

To find the last number (the unit digit) in the expression \( 4 + 9^2 + 4^3 + 9^4 + 4^5 + 9^6 + \ldots + 4^{99} + 9^{100} \), we will focus on the unit digits of each term in the expression.

### Step-by-Step Solution:

1. **Identify the pattern in the unit digits of powers of 4:**
- \( 4^1 = 4 \) (unit digit is 4)
- \( 4^2 = 16 \) (unit digit is 6)
- \( 4^3 = 64 \) (unit digit is 4)
- \( 4^4 = 256 \) (unit digit is 6)
- The unit digits of powers of 4 alternate between 4 and 6.
- For odd powers of 4, the unit digit is 4, and for even powers, it is 6.

2. **Identify the pattern in the unit digits of powers of 9:**
- \( 9^1 = 9 \) (unit digit is 9)
- \( 9^2 = 81 \) (unit digit is 1)
- \( 9^3 = 729 \) (unit digit is 9)
- \( 9^4 = 6561 \) (unit digit is 1)
- The unit digits of powers of 9 also alternate between 9 and 1.
- For odd powers of 9, the unit digit is 9, and for even powers, it is 1.

3. **List the unit digits for the entire expression:**
- The expression alternates between terms of \( 4^n \) and \( 9^n \):
- \( 4^1 \) (4), \( 9^2 \) (1), \( 4^3 \) (4), \( 9^4 \) (1), \( 4^5 \) (4), \( 9^6 \) (1), ..., up to \( 4^{99} \) and \( 9^{100} \).
- The unit digits for odd \( n \) (for \( 4^n \)): 4 (for all odd \( n \)).
- The unit digits for even \( n \) (for \( 9^n \)): 1 (for all even \( n \)).

4. **Count the number of terms:**
- The total number of terms from \( 4^1 \) to \( 9^{100} \) is 100 terms.
- There are 50 odd terms (where \( n \) is odd) and 50 even terms (where \( n \) is even).

5. **Calculate the sum of the unit digits:**
- The sum of unit digits for odd \( n \) (50 terms of 4): \( 50 \times 4 = 200 \).
- The sum of unit digits for even \( n \) (50 terms of 1): \( 50 \times 1 = 50 \).
- Total sum of unit digits = \( 200 + 50 = 250 \).

6. **Find the unit digit of the total sum:**
- The unit digit of 250 is 0.

### Conclusion:
The last number (unit digit) in the expression \( 4 + 9^2 + 4^3 + 9^4 + \ldots + 4^{99} + 9^{100} \) is **0**.