Question

Alguém pode ajudar ??

Answer 1

O valor de K que torna verdadeira a igualdade é K = 2,0 ; isto é, o item B) está correto.

$\large\begin{array}{l}\!\displaystyle\sf\sum_{i\,=\:\!1}^{15}(i+3)=\sum_{i\,=\,1}^{15}(2K)+\sum_{i\,=\,6}^{15}i\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{15}(i+3)=(2K)\cdot 15+\sum_{i\,=\:\!1}^{15}i-\sum_{i\,=\:\!1}^{5}i\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{15}(i+3)-\sum_{i\,=\:\!1}^{15}i=30^{}K-\underbrace{\sf (1+2+3+4+5)}_{15}\end{array}$

$\large\begin{array}{l}\displaystyle\sf\sum_{i\,=\:\!1}^{15}\big[(i+3)-i\big]=30K-15\\\\ \displaystyle\sf\sum_{i\,=\:\!1}^{15}\sf 3=30K-15\\\\ \sf 3\cdot 15=\sf 30K-15\\\\ \sf 30K-15=45\\\\ \sf 30K=45+15\\\\ \sf 30K=60\\\\ \sf K=\dfrac{60}{30}\\\\ \boxed{\begin{array}{l}\sf K=2\end{array}}\end{array}$