Define the original state

The initial mean is given as $20$. Let the observations be $x_1, x_2,..., x_{10}$. The formula for mean is $\text{Mean} = \frac{\sum x_i}{n}$.

$\text{Mean} = \frac{\sum_{i=1}^{10} x_i}{10} = 20$

Apply the modification

If every observation is increased by $5$, the new observations become $(x_1 + 5), (x_2 + 5),..., (x_{10} + 5)$.

$\text{New Sum} = \sum_{i=1}^{10} (x_i + 5) = \sum x_i + (10 \times 5)$

Calculate the new mean

Divide the new sum by the number of observations $10$: $\text{New Mean} = \frac{\sum x_i + 50}{10} = \frac{\sum x_i}{10} + 5$.

$\text{New Mean} = 20 + 5 = 25$