Đáp án:
Giải thích các bước giải:
`A=1+4+4^2+4^3+...+4^2020+4^2021`
`=>4A=4.(1+4+4^2+4^3+...+4^2020+4^2021)`
`=>4A=4+4^2+4^3+4^4+...+4^2021+4^2022`
`=>4A-A=(4+4^2+4^3+4^4+...+4^2021+4^2022)-(1+4+4^2+4^3+...+4^2020+4^2021)`
`=>3A=4^2022-1`
`=>3A+1=4^2022-1+1`
`=>3A+1=4^2022`
`=>3A+1=2^4044=2^n`
`=>2^4044=2^n`
`=>n=4044`.
Vậy `n=4044`.
b) S`=1.2+2.3+3.4+...+99.100`
`=>`3S`=1.2.3+2.3.3+3.4.3+...+99.100.3`
`=>`3S`=1.2.3+2.3.(4-1)+3.4.(5-2)+...+99.100.(101-98)`
`=>`3S`=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100`
`=>`3S`=(1.2.3+2.3.4+3.4.5+...+99.100.101)-(1.2.3+2.3.4+...+98.99.100)`
`=>`3S`=99.100.101`
`=>`S`=33.100.101`
`=>`S`=333300`
Vậy `S=333300`.
c) `2^x+2^(x+1)+2^(x+2)+...+2^(x+2015)=2^2017-2`
`=>2^x.(1+2+2^2+...+2^2015)=2^2017-2`
`=>A=1+2+2^2+...+2^2015`
`=>2A=2.(1+2+2^2+...+2^2015)`
`=>2A=2+2^2+2^3+...+2^2016`
`=>2A-A=(2+2^2+2^3+...+2^2016)-(1+2+2^2+...+2^2015)`
`=>A=2^2016-1`
`=>2^x.(2^2016-1)=2^2017-2`
`=>2^x=(2^2017-2)/(2^2016-1)`
`=>2^x=(2.(2^2016-1))/(2^2016-1`
`=>2^x=2`
`=>x=1`
Vậy `x=1`.
