Question 1210226
Let's analyze how the numbers in the sequence are formed by sums of distinct powers of 5:

Powers of 5 are:
$5^0 = 1$
$5^1 = 5$
$5^2 = 25$
$5^3 = 125$
and so on.

The sequence consists of numbers that can be written in the form $c_0 \cdot 5^0 + c_1 \cdot 5^1 + c_2 \cdot 5^2 + c_3 \cdot 5^3 + \dots$, where each $c_i$ is either 0 or 1 (because the powers of 5 must be distinct).

Let's generate the terms of the sequence in increasing order:

Using only $5^0 = 1$:
$1 \cdot 5^0 = 1$

Using $5^0 = 1$ and $5^1 = 5$:
$1 \cdot 5^0 = 1$ (already listed)
$1 \cdot 5^1 = 5$
$1 \cdot 5^0 + 1 \cdot 5^1 = 1 + 5 = 6$

Using $5^0 = 1$, $5^1 = 5$, and $5^2 = 25$:
$1 \cdot 5^0 = 1$
$1 \cdot 5^1 = 5$
$1 \cdot 5^2 = 25$
$1 \cdot 5^0 + 1 \cdot 5^1 = 1 + 5 = 6$
$1 \cdot 5^0 + 1 \cdot 5^2 = 1 + 25 = 26$
$1 \cdot 5^1 + 1 \cdot 5^2 = 5 + 25 = 30$
$1 \cdot 5^0 + 1 \cdot 5^1 + 1 \cdot 5^2 = 1 + 5 + 25 = 31$

The sequence generated so far in increasing order is: 1, 5, 6, 25, 26, 30, 31, ...

We are looking for the first term that is greater than 50. Let's continue generating terms by including the next power of 5, which is $5^3 = 125$.

The terms formed using $5^0, 5^1, 5^2$ are all less than or equal to $1 + 5 + 25 = 31$. Now, let's consider sums that include $5^3 = 125$:

$1 \cdot 5^3 = 125$
$1 \cdot 5^0 + 1 \cdot 5^3 = 1 + 125 = 126$
$1 \cdot 5^1 + 1 \cdot 5^3 = 5 + 125 = 130$
$1 \cdot 5^2 + 1 \cdot 5^3 = 25 + 125 = 150$
$1 \cdot 5^0 + 1 \cdot 5^1 + 1 \cdot 5^3 = 1 + 5 + 125 = 131$
$1 \cdot 5^0 + 1 \cdot 5^2 + 1 \cdot 5^3 = 1 + 25 + 125 = 151$
$1 \cdot 5^1 + 1 \cdot 5^2 + 1 \cdot 5^3 = 5 + 25 + 125 = 155$
$1 \cdot 5^0 + 1 \cdot 5^1 + 1 \cdot 5^2 + 1 \cdot 5^3 = 1 + 5 + 25 + 125 = 156$

Looking at the sequence 1, 5, 6, 25, 26, 30, 31, ..., the next terms will be formed by sums of distinct powers of 5 greater than 31. The next power of 5 is 125. The smallest positive integer that can be expressed as a sum of distinct powers of 5 and is greater than 31 will involve the smallest power of 5 that is greater than 31, which is 125.

The smallest term in the sequence that includes $5^3 = 125$ is $1 \cdot 5^3 = 125$.

Since 125 is greater than 50, and all previous terms in the sequence (formed by sums of 1, 5, and 25) are less than or equal to $1+5+25=31$, the first term in the sequence that is greater than 50 is 125.

Final Answer: The final answer is $\boxed{125}$