Question

va rog repede. dau coroana!

Answer 1

   
[tex]\displaystyle\\ \texttt{c)}\\ A=2^2+2^4+2^6+\cdots +2^{30}\\\\ \text{Observam ca suma primilor doi termeni = 20}\\\\ 2^2+2^4 = 4 + 16 = 20\\\\ \text{Trebuie sa grupam sirul in grupe de cate 2 termeni.}\\\\ \text{Verificam daca avem voie, calculand numarul de termeni:}\\\\ n = \frac{30 -2}{2} + 1 = \frac{28}{2} + 1 = 14+1 = 15~termeni.\\\\ \text{15 nu este divizibil cu 2.}\\\\ \Longrightarrow~~\boxed{\text{A nu este divizibil cu 20}}[/tex]


[tex]\displaystyle\\ \texttt{d)}\\\\ A=1+5^2+5^4+\cdots+5^{34}=5^0+5^2+5^4+\cdots+5^{34}\\\\ \text{Observam ca suma primilor doi termeni=26}\\\\ 5^0+5^2=1+25=26\\\\ \text{Trebuie sa grupam sirul in grupe de cate 2 termeni.}\\\\ \text{Verificam daca avem voie, calculand numarul de termeni:}\\\\ n = \frac{34 -0}{2} + 1 = \frac{34}{2} + 1 = 17+1 = 18~termeni.\\\\ \text{18 este divizibil cu 2, deci putem grupa sirul in grupe de 2 termeni.} [/tex]

[tex]\displaystyle\\ A = (5^0 + 5^2)+(5^4+5^6)+\cdots+(5^3+5^{34}) =\\\\ = (5^0 + 5^2)+5^4(5^0+5^2)+\cdots+ 5^{32}(5^{0}+5^{2}) =\\\\ (5^0 + 5^2)(1+5^4+\cdots + 5^{32}) =\boxed{26(1+5^4+\cdots + 5^{32}) ~\vdots~26} [/tex]


[tex]\displaystyle\\ \texttt{e)}\\\\ A = 7^1+7^2+7^3+7^4+7^5+7^6+7^7+7^8+\cdots+7^{97}+7^{98}+7^{99}+7^{100}\\\\ \text{Calculam suma primilor 4 termeni:}\\\\ 7^1+7^2+7^3+7^4=7+49 + 343+2401=2800~\vdots~50\\\ \text{Vom imparti siul in grupe de cate 4 termeni.}\\\\ \text{Avem voie deoarece avem 100 de termeni si 100 se divide cu 4.}\\\\ (7^1+7^2+7^3+7^4)+(7^5+7^6+7^7+7^8)+\cdots+\\ +(7^{97}+7^{98}+7^{99}+7^{100})=\\\\ (7^1+7^2+7^3+7^4)+7^4(7^1+7^2+7^3+7^4)+\cdots+\\ +7^{96}(7^1+7^2+7^3+7^4)=\\\\ [/tex]

[tex]\displaystyle\\ (7^1+7^2+7^3+7^4)+7^4(7^1+7^2+7^3+7^4)+\cdots+\\ +7^{96}(7^1+7^2+7^3+7^4)=\\\\ =(7^1+7^2+7^3+7^4)(1+7^4+\cdots+7^{96}) =\\ = \boxed{2800(1+7^4+\cdots+7^{96})~\vdots~50}[/tex]