Question

Aflati x
2^{x} + 2^{x+1} + 2^{x+2} =56
2+2^2+2^3+...+2^x=2^55-2

Answer 1

[tex]a)~2^x+2^{x+1}+2^{x+2}=56 \\ 2^x+2^x\cdot2^1+2^x\cdot2^2=56 \\ 2^x(1+2+4)=56 \\ 2^x\cdot7=56 \\ 2^x=8 \\ 2^x=2^3\rightarrow x=3 \\ \\ b)1+a+a^2+a^3+...+a^n= \frac{a^{n+1}-1}{a-1} \\ a+a^2+a^3+ ...+a^n= \frac{a^{n+1}-1}{a-1}-1 \\ 2+2^2+2^3+...+2^x= \frac{2^{x+1}-1}{2-1}-1 \\ 2+2^2+2^3+...+2^x= \frac{2^{x+1}-1}{1}-1 \\ 2+2^2+2^3+...+2^x=2^{x+1}-1-1 \\ 2+2^2+2^3+...+2^x=2^{x+1}-2 \\ 2+2^2+2^3+...+2^x=2^5^5-2 \\ 2^{x+1}-2=2^5^5-2\rightarrow x+1=55\rightarrow x=54 [/tex]