Concept:Understanding how the mean changes when a constant is added to each observation, and how the sum of an arithmetic progression is used.Explanation:We are given 10 observations: $x_1, x_2, \dots, x_{10}$.The mean of these observations is given as 30.The formula for the mean is: Mean = $\frac{\text{Sum of observations}}{\text{Number of observations}}$So, for the first case:30 = $\frac{x_1 + x_2 + \dots + x_{10}}{10}$Multiplying both sides by 10, we get the sum of the original observations:$x_1 + x_2 + \dots + x_{10} = 30 \times 10 = 300$.Now, consider the new set of observations: $x_1 + 2, x_2 + 4, x_3 + 6, \dots, x_{10} + 20$.Notice that the numbers added to each $x_i$ form an arithmetic progression: 2, 4, 6, ..., 20.The terms added are $2 \times 1, 2 \times 2, 2 \times 3, \dots, 2 \times 10$.The sum of the new observations is:( $x_1 + 2$) + ($x_2 + 4$) + ($x_3 + 6$) + $\dots$ + ($x_{10} + 20$)We can rearrange this sum as:( $x_1 + x_2 + \dots + x_{10}$) + ( $2 + 4 + 6 + \dots + 20$ )We already know that $x_1 + x_2 + \dots + x_{10} = 300$.Now, let's find the sum of the arithmetic progression $2 + 4 + 6 + \dots + 20$.This is an arithmetic series with the first term $a = 2$, the last term $l = 20$, and the number of terms $n = 10$.The sum of an arithmetic series is given by $S_n = \frac{n}{2}(a + l)$.So, the sum of the added terms is: $\frac{10}{2}(2 + 20) = 5 \times 22 = 110$.Therefore, the sum of the new observations is $300 + 110 = 410$.The mean of the new observations is:New Mean = $\frac{\text{Sum of new observations}}{\text{Number of observations}}$New Mean = $\frac{410}{10}$New Mean = 41.Answer:41