Let's go through the problem step by step. The given data are $x_1 = 2$, $x_2 = 5$, $x_3 = 3$, and $x_4 = 4$. We are asked to calculate the following sums:

a) $\sum_{i=1}^4 x_i$

This sum is just the sum of all the given $x_i$: $\sum_{i=1}^4 x_i = x_1 + x_2 + x_3 + x_4 = 2 + 5 + 3 + 4 = 14$

b) $\sum_{i=1}^4 8$

This is a constant sum where each term is 8: $\sum_{i=1}^4 8 = 8 + 8 + 8 + 8 = 32$

c) $\sum_{i=1}^4 2x_i$

Here, each term is multiplied by 2: $\sum_{i=1}^4 2x_i = 2(x_1) + 2(x_2) + 2(x_3) + 2(x_4) = 2(2) + 2(5) + 2(3) + 2(4) = 4 + 10 + 6 + 8 = 28$

d) $\sum_{i=1}^4 (x_i - 3)$

This involves subtracting 3 from each $x_i$: $\sum_{i=1}^4 (x_i - 3) = (2 - 3) + (5 - 3) + (3 - 3) + (4 - 3) = (-1) + 2 + 0 + 1 = 2$

e) $\sum_{i=1}^4 x_i^2$

Here we square each $x_i$: $\sum_{i=1}^4 x_i^2 = 2^2 + 5^2 + 3^2 + 4^2 = 4 + 25 + 9 + 16 = 54$

f) $\sum_{i=1}^4 (x_i - 3)^2$

We first subtract 3 from each $x_i$ and then square the result: $\sum_{i=1}^4 (x_i - 3)^2 = (2 - 3)^2 + (5 - 3)^2 + (3 - 3)^2 + (4 - 3)^2 = (-1)^2 + 2^2 + 0^2 + 1^2 = 1 + 4 + 0 + 1 = 6$

Summary of the results:

• a) $\sum_{i=1}^4 x_i = 14$
• b) $\sum_{i=1}^4 8 = 32$
• c) $\sum_{i=1}^4 2x_i = 28$
• d) $\sum_{i=1}^4 (x_i - 3) = 2$
• e) $\sum_{i=1}^4 x_i^2 = 54$
• f) $\sum_{i=1}^4 (x_i - 3)^2 = 6$

Would you like further details or explanations on any step?

Here are five related questions to deepen your understanding:

1. What is the general form of a summation and how can it be simplified?
2. How do constant sums like in part (b) affect the total sum?
3. What is the importance of squaring terms in a summation?
4. How would the results change if different values for $x_i$ were provided?
5. Can you think of real-world scenarios where such summations are useful?

Tip: When solving summations, simplify step by step to avoid errors, especially when variables or constants are involved.