Thu gọn biểu thức A

To simplify the expression \( A = 1 + \frac{1}{5} + \frac{1}{5^2} + \frac{1}{5^3} + \ldots + \frac{1}{5^{2024}} \), we can recognize this as a geometric series.

The formula for the sum of a geometric series is:

\[
S_n = a \frac{1 - r^n}{1 - r}
\]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term,
- \( r \) is the common ratio,
- \( n \) is the number of terms.

For our series:
- \( a = 1 \)
- \( r = \frac{1}{5} \)
- \( n = 2025 \) (since we have terms from \( 0 \) to \( 2024 \))

Now we apply the formula:

\[
A = 1 \cdot \frac{1 - \left(\frac{1}{5}\right)^{2025}}{1 - \frac{1}{5}} = \frac{1 - \left(\frac{1}{5}\right)^{2025}}{\frac{4}{5}}
\]

Simplifying this expression gives:

\[
A = \frac{5}{4} \left(1 - \left(\frac{1}{5}\right)^{2025}\right)
\]

So the simplified form of \( A \) is:

\[
A = \frac{5}{4} \left( 1 - \frac{1}{5^{2025}} \right)
\]