A=3^2+3^4+3^6+...+3^100 prove it is not divisible by 4

The given sum is \( A = 3^2 + 3^4 + 3^6 + \dots + 3^{100} \). We want to prove that \( A \) is not divisible by 4.

### Step-by-Step Solution:

1. **General Form of the Series**:  
   The sum is a geometric series where the exponents are all even powers of 3. The general term of this series is \( 3^{2n} \), where \( n \) is an integer starting from 1 and goes up to 50. Thus, the sum can be rewritten as:
   \[
   A = 3^2 + 3^4 + 3^6 + \dots + 3^{100}
   \]

2. **Modulo 4 Analysis of Powers of 3**:  
   First, we investigate the behavior of powers of 3 modulo 4.  
   Start by calculating a few values of \( 3^n \mod 4 \):
   - \( 3^1 \equiv 3 \mod 4 \)
   - \( 3^2 = 9 \equiv 1 \mod 4 \)
   - \( 3^3 = 27 \equiv 3 \mod 4 \)
   - \( 3^4 = 81 \equiv 1 \mod 4 \)
   - Continuing, we see that:
     \[
     3^{2n} \equiv 1 \mod 4 \quad \text{for all } n.
     \]
   This is because \( 3^2 \equiv 1 \mod 4 \) and the powers of \( 3^2 \) repeat this cycle.

3. **Sum of the Series Modulo 4**:  
   Now, we apply the modulo 4 result to the entire sum. Since every term in the sum \( 3^{2n} \equiv 1 \mod 4 \), each term contributes 1 modulo 4.  
   There are 50 terms in the series (from \( 3^2 \) to \( 3^{100} \)). Therefore, the sum is:
   \[
   A \equiv 1 + 1 + 1 + \dots + 1 \mod 4
   \]
   This is equivalent to adding 1 fifty times:
   \[
   A \equiv 50 \cdot 1 \equiv 50 \mod 4
   \]
   Now, calculate \( 50 \mod 4 \):
   \[
   50 \mod 4 = 2
   \]

4. **Conclusion**:  
   Since \( A \equiv 2 \mod 4 \), the sum \( A \) is **not divisible by 4**.

Would you like further details or have any questions?

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### Related Questions:
1. How would the sum behave if the exponents were odd?
2. Can we generalize this proof for any powers of 3?
3. What would change if we calculated the sum modulo 5 instead of 4?
4. What if the sum was from \( 3^1 \) to \( 3^{99} \) instead of even exponents?
5. How does the geometric series sum formula apply here for a different modulo?

### Tip:
When working with powers in modular arithmetic, observe patterns in the sequence of powers, as they often repeat cyclically.

A = 3^2 + 3^4 + 3^6 + ... + 3^100 prove it is not divisible by 4.

Explore a detailed step-by-step proof to show that the sum A = 3^2 + 3^4 + ... + 3^100 is not divisible by 4. Learn about modular arithmetic, cyclic patterns in powers of 3 modulo 4, and geometric series. Suitable for Grades 10-12.

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