Đáp án:
Giải thích các bước giải:
a) C=2+$2^{2}$ +$2^{3}$+...+$2^{99}$+$2^{100}$
C = (2+$2^{2}$ +$2^{3}$+$2^{4}$+$2^{5}$)+(2+$2^{6}$ +$2^{7}$+$2^{8}$+$2^{9}$+$2^{10}$)+...+($2^{96}$+$2^{97}$+$2^{98}$+$2^{99}$+$2^{100}$)
C = 62+$2^{4}$.(2+$2^{2}$ +$2^{3}$+$2^{4}$+$2^{5}$)+...+$2^{95}$.(2+$2^{2}$ +$2^{3}$+$2^{4}$+$2^{5}$)
C = 62+$2^{4}$.62+...+$2^{95}$.62
C = 62.(1+$2^{4}$+...+$2^{95}$)=31.2.(1+$2^{4}$+...+$2^{95}$) chia hết cho 31. (đpcm)
b) C=2+$2^{2}$ +$2^{3}$+...+$2^{99}$+$2^{100}$
⇒2C = $2^{2}$ +$2^{3}$+$2^{4}$+...+$2^{99}$+$2^{100}$+$2^{101}$
⇒2C - C = ($2^{2}$ +$2^{3}$+$2^{4}$+...+$2^{99}$+$2^{100}$+$2^{101}$)-(2+$2^{2}$ +$2^{3}$+...+$2^{99}$+$2^{100}$)
⇒C = $2^{101}$ - 2
Thay vào phép tính, ta có: $2^{2x-1}$-2=$2^{101}$ - 2
⇒ $2^{2x-1}$=$2^{101}$
⇒ 2x-1=101
2x =101-1
2x =100
x =100:2
x =50
Chúc bạn thi tốt!
