Questions: Resolva os somatórios: i X i 1 5 2 9 3 -4 4 -2 5 1 6 0 7 12 a) sum from i=1 to 3 of (xi) b) sum from i=1 to 7 of (xi)^2-1 c) sum from i=1 to 5 of (xi+8)' d) sum from i=1 to 3 of (sqrt(xi)) e) sum from i=1 to 6 of (xi / 12)

Resolva os somatórios:

i X i
1 5
2 9
3 -4
4 -2
5 1
6 0
7 12

a) sum from i=1 to 3 of (xi)
b) sum from i=1 to 7 of (xi)^2-1
c) sum from i=1 to 5 of (xi+8)'
d) sum from i=1 to 3 of (sqrt(xi))
e) sum from i=1 to 6 of (xi / 12)

Transcript text: Resolva os somatórios: \begin{tabular}{|c|c|} \hline $\mathbf{i}$ & $\mathbf{X i}$ \\ \hline 1 & 5 \\ \hline 2 & 9 \\ \hline 3 & -4 \\ \hline 4 & -2 \\ \hline 5 & 1 \\ \hline 6 & 0 \\ \hline 7 & 12 \\ \hline \end{tabular} a) $\sum_{i=1}^{3}(x i)$ b) $\sum_{i=1}^{7}(x i)^{2}-1$ c) $\sum_{i=1}^{5}(x i+8)^{\prime}$ d) $\sum_{i=1}^{3}(\sqrt{x i})$ e) $\sum_{i=1}^{6}(x i / 12)$

Solution

Solution Steps

To solve the given summation problems, we will follow these approaches:

a) For $\sum_{i=1}^{3}(x_i)$, sum the values of $x_i$ from $i=1$ to $i=3$.

b) For $\sum_{i=1}^{7}(x_i)^{2}-1$, calculate the square of each $x_i$ from $i=1$ to $i=7$, sum them up, and then subtract 1 from the total.

c) For $\sum_{i=1}^{5}(x_i+8)$, add 8 to each $x_i$ from $i=1$ to $i=5$ and sum the results.

Step 1: Calculate $\sum_{i=1}^{3}(x_i)$

To find $\sum_{i=1}^{3}(x_i)$, we sum the values of $x_i$ for $i = 1$ to $3$. Given the values $x_1 = 5$, $x_2 = 9$, and $x_3 = -4$, we have:

\[ \sum_{i=1}^{3}(x_i) = 5 + 9 + (-4) = 10 \]

Step 2: Calculate $\sum_{i=1}^{7}(x_i)^2 - 1$

First, calculate $(x_i)^2$ for each $i$ from $1$ to $7$, then sum these values and subtract $1$.

\[ \begin{align_} x_1^2 &= 5^2 = 25, \\ x_2^2 &= 9^2 = 81, \\ x_3^2 &= (-4)^2 = 16, \\ x_4^2 &= (-2)^2 = 4, \\ x_5^2 &= 1^2 = 1, \\ x_6^2 &= 0^2 = 0, \\ x_7^2 &= 12^2 = 144. \end{align_} \]

Sum these values:

\[ \sum_{i=1}^{7}(x_i)^2 = 25 + 81 + 16 + 4 + 1 + 0 + 144 = 271 \]

Subtract $1$:

\[ \sum_{i=1}^{7}(x_i)^2 - 1 = 271 - 1 = 270 \]

Step 3: Calculate $\sum_{i=1}^{5}(x_i + 8)$

Add $8$ to each $x_i$ for $i = 1$ to $5$ and sum the results:

\[ \begin{align_} (x_1 + 8) &= 5 + 8 = 13, \\ (x_2 + 8) &= 9 + 8 = 17, \\ (x_3 + 8) &= -4 + 8 = 4, \\ (x_4 + 8) &= -2 + 8 = 6, \\ (x_5 + 8) &= 1 + 8 = 9. \end{align_} \]

Sum these values:

\[ \sum_{i=1}^{5}(x_i + 8) = 13 + 17 + 4 + 6 + 9 = 49 \]