1×5/3+5×9/3+9×13/3+…+1997×2001/3=_____.
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1×5/3+5×9/3+9×13/3+…+1997×2001/3=_____.
请大家帮我算算,
注:"/"为分数线.
例:1×5/3读作:一乘五分之三.
3
-----
1×5
请大家帮我算算,
注:"/"为分数线.
例:1×5/3读作:一乘五分之三.
3
-----
1×5
易知此数列的通项为
an = 3/[(4n-3)(4n+1)] = 3/4[1/(4n-3) - 1/(4n+1)]
所以
S = 3/(1*5) +3/(5*9)+3/(9*13)+.+3/(1997*2001)
= 3/4(1-1/5) + 3/4(1/5-1/9)+3/4(1/9-1/13)+.+3/4(1/1997-1/2001)
= 3/4[(1-1/5)+(1/5-1/9)+(1/9-1/13)+.+(1/1997-1/2001)]
= (3/4)*(1-1/2001)
= 500/667
an = 3/[(4n-3)(4n+1)] = 3/4[1/(4n-3) - 1/(4n+1)]
所以
S = 3/(1*5) +3/(5*9)+3/(9*13)+.+3/(1997*2001)
= 3/4(1-1/5) + 3/4(1/5-1/9)+3/4(1/9-1/13)+.+3/4(1/1997-1/2001)
= 3/4[(1-1/5)+(1/5-1/9)+(1/9-1/13)+.+(1/1997-1/2001)]
= (3/4)*(1-1/2001)
= 500/667
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