It can be easily observable that an equation where both side are sum of two number with same base but different power is not possible unless the base is $-1, 0, 1, 2$
So we will consider the powers are same.
In the equation
$5^{2r+1} + 5^{2} = 5^{r} + 5^{r+3}$
We will split the equation into two case: Case 1: $2r+1 = r+3$ and $ 2=r$
This two equation can be true at the same time and here $r = 2$
Case 2: $2r + 1 = r$ and $2 = r+3$
This two equation can also be true at the same time and here $r = -1$
So the solution of $r = (2,-1)$
And their sum $= (2) + (-1) = \boxed{1}$
And if there is any error , then inform me. A solution without using algebra
It can be easily observable that an equation where both side are sum of two number with same base but different power is not possible unless the base is $-1, 0, 1, 2$
So we will consider the powers are same.
In the equation
$5^{2r+1} + 5^{2} = 5^{r} + 5^{r+3}$
We will split the equation into two case: Case 1: $2r+1 = r+3$ and $ 2=r$
This two equation can be true at the same time and here $r = 2$
Case 2: $2r + 1 = r$ and $2 = r+3$
This two equation can also be true at the same time and here $r = -1$
So the solution of $r = (2,-1)$
And their sum $= (2) + (-1) = \boxed{1}$
And if there is any error , then inform me. A solution without using algebra
