Question

Questão! alguém ?? verifique o resultado dos seguintes somatórios:

Answer 1

Nos itens a) e b), os valores solicitados são – 8 e 22, respectivamente. Item correto: A).

Letra a)

Dado que

$\large\begin{cases}\displaystyle\sf\sum_{i\,=\,1}^{5}X_i=6\\\\ \sf\ e\\\\ \displaystyle\sf\sum_{i\,=\,1}^{5}\big(X_i\big)^{\!\:\!2}=42\end{cases}$

, temos:

$\large\begin{array}{l}\!\:\!\displaystyle\sf\sum_{i\,=\:\!1}^{5}\big(2\:\!X_i-4\big)=\sum_{i\,=\,1}^{5}\big(2\:\!X_i\big)-\sum_{i\,=\:\!1}^{5}4\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{5}\big(2\:\!X_i-4\big)=\:\!2\cdot\!\underbrace{\sf \sum_{i\,=\:\!1}^{5}X_i}_{6}\:\!-\:4\cdot5\\ \displaystyle\sf\sum_{i\,=\:\!1}^{5}\big(2\:\!X_i-4\big)=\:\!2\cdot 6-4\cdot 5\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{5}\big(2\:\!X_i-4\big)=\:\!12-20\\\\ \boxed{\displaystyle\sf\sum_{i\,=\,1}^{5}\big(2\:\!X_i-4\big)=-\,8}\end{array}$

Letra b)

Usando as informações apresentadas no item a), encontraremos:

$\large\begin{array}{l}\!\displaystyle\sf\sum_{i\,=\:\!1}^{5}\big[\big(X_i-2\big)\!\:\!\big(X_i+2\big)\big]=\sum_{i\,=\:\!1}^{5}\big[\big(X_i\big)^{\!\:\!2}\!\:\!-\,2^2\big]\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{5}\big[\big(X_i-2\big)\!\:\!\big(X_i+2\big)\big]=\underbrace{\sf\sum_{i\,=\:\!1}^{5}\big(X_i\big)^{\!\:\!2}}_{42}-\sum_{i\,=\:\!1}^{5}4\end{array}$

$\large\begin{array}{l}\!\displaystyle\sf\sum_{i\,=\:\!1}^{5}\big[\big(X_i-2\big)\!\:\!\big(X_i+2\big)\big]=\:\!42-4\cdot 5\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{5}\big[\big(X_i-2\big)\!\:\!\big(X_i+2\big)\big]=\:\!42-20\\\\ \boxed{\displaystyle\sf\sum_{i\,=\:\!1}^{5}\big[\big(X_i-2\big)\!\:\!\big(X_i+2\big)\big]=\:\!22}\end{array}$