Lời giải.
Bài `1.`
`a)`
`S=6+6^2+6^3+6^4+...+6^99`
`6S=6.(6+6^2+6^3+6^4+...+6^99)`
`6S=6.6+6.6^2+6.6^3+6.6^4+...+6.6^99`
`6S=6^2+6^3+6^4+6^5+...+6^100`
Suy ra, `6S-S=6^2+6^3+6^4+6^5+...+6^100-6-6^2-6^3-6^4-...-6^99`
`5S=(6^2-6^2)+(6^3-6^3)+...+(6^99-6^99)+(6^100-6)`
`5S=0+0+...+0+(6^100-6)`
Suy ra `S={6^100-6}/5.`
`b)`
`S=1+4+4^2+4^3+...+4^100`
`4S=4.(1+4+4^2+4^3+...+4^100)`
`4S=4.1+4.4+4.4^2+4.4^3+...+4.4^100`
`4S=4+4^2+4^3+4^4+...+4^101`
Suy ra, `4S-S=(4+4^2+4^3+4^4+...+4^101)-(1+4+4^2+4^3+...+4^100)`
`3S=4+4^2+4^3+4^4+...+4^101-1-4-4^2-4^3-...-4^100`
`3S=(4-4)+(4^2-4^2)+(4^3-4^3)+(4^4-4^4)+...+(4^100-4^100)+4^101-1`
`3S=0+0+0+0+...+0+4^101-1`
`3S=4^101-1`
`S={4^101-1}/3.`
`c)`
`S=1+1/2+1/2^2+1/2^3+1/2^4+...+1/2^99+1/2^100`
`1/2 .S=1/2 . (1+1/2+1/2^2+1/2^3+1/2^4+...+1/2^99+1/2^100)`
`1/2 . S= 1/2 + 1/2^2 + 1/2^3 + 1/2^4 +... + 1/2^100 + 1/2^101`
Suy ra `S- 1/2S =(1+1/2+1/2^2+1/2^3+1/2^4+...+1/2^99+1/2^100)-(1/2 + 1/2^2 + 1/2^3 + 1/2^4 +... + 1/2^100 + 1/2^101)`
`1/2S= (1-1/2^101) + (1/2 - 1/2) +(1/2^2-1/2^2) +... +( 1/2^100-1/2^100)`
`1/2S=1-1/2^101`
Suy ra `2. 1/2S= 2. (1-1/2^101)`
`S=2-2/2^101=2-1/2^100.`
`d)`
`S=1/3+1/3^2+1/3^3+1/3^4+...+1/3^99+1/3^100`
`1/3 .S=1/3 . (1/3+1/3^2+1/3^3+1/3^4+...+1/3^99+1/3^100)`
`1/3 . S= 1/3^2 + 1/3^3 + 1/3^4 +... + 1/3^100 + 1/3^101`
Suy ra `S- 1/3S =(1/3+1/3^2+1/3^3+1/3^4+...+1/3^99+1/3^100)-( 1/3^2 + 1/3^3 + 1/3^4 +... + 1/3^100 + 1/3^101)`
`2/3S=(1/3- 1/3^101 )+ (1/3^2 - 1/3^2) +(1/2^2-1/2^2) +... +( 1/3^100-1/3^100)`
`2/3S= 1/3 - 1/3^101`
Suy ra `3/2. 2/3S= 3/2. (1/3 - 1/3^101)`
`S=1/2-3/{2. 3^101}= 1/2 - 1/{2. 3^100}.`
Bài `2.`
`A={2^30 . 5^7 + 2^13 . 5^27}/ {2^27 . 5^7 + 2^10 . 5^27}`
`A={2^13 . 2^17 . 5^7 + 2^13 . 5^7 . 5^20}/ {2^10 . 2^17 . 5^7 + 2^10 . 5^7 . 5^20}`
`A={2^13 .5^7 . (2^17 + 5^20)}/ {2^10 . 5^7 . (2^17+5^20)}`
`A={2^13}/{2^10}=2^3=8.`
