已知f(x)=x^3+999 求f(50)+f(51)+f(52)+……+f(100)等于多少?
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已知f(x)=x^3+999 求f(50)+f(51)+f(52)+……+f(100)等于多少?
已知f(x)=x^3+999 求f(50)+f(51)+f(52)+……+f(100)等于多少?
已知f(x)=x^3+999 求f(50)+f(51)+f(52)+……+f(100)等于多少?
1^3+2^3+……+n^3=[n(n+1)/2]^2
所以1^3+2^3+……+100^3=[100*101/2]^2=5050^2
1^3+2^3+……+49^3=[49*50/2]^2=1225^2
所以50^3+51^3+……+100^3=5050^2-1225^2
f(50)+f(51)+f(52)+……+f(100)
=50^3+51^3+……+100^3+51*999
=5050^2-1225^2+51*999
=24052824
1^3+2^3+.....+n^3=n^2(n+1)^2/4=[n(n+1)/2]^2
推导过程:
(n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2]
=(2n^2+2n+1)(2n+1)
=4n^3+6n^2+4n+1
2^4-1^4=4*1^3+6*1^2+4*1+1
3^4-2^4=4*2^3+6*2^2+...
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1^3+2^3+.....+n^3=n^2(n+1)^2/4=[n(n+1)/2]^2
推导过程:
(n+1)^4-n^4=[(n+1)^2+n^2][(n+1)^2-n^2]
=(2n^2+2n+1)(2n+1)
=4n^3+6n^2+4n+1
2^4-1^4=4*1^3+6*1^2+4*1+1
3^4-2^4=4*2^3+6*2^2+4*2+1
4^4-3^4=4*3^3+6*3^2+4*3+1
......
(n+1)^4-n^4=4*n^3+6*n^2+4*n+1
各式相加有
(n+1)^4-1=4*(1^3+2^3+3^3...+n^3)+6*(1^2+2^2+...+n^2)+4*(1+2+3+...+n)+n
4*(1^3+2^3+3^3+...+n^3)=(n+1)^4-1+6*[n(n+1)(2n+1)/6]+4*[(1+n)n/2]+n
=[n(n+1)]^2
1^3+2^3+...+n^3=[n(n+1)/2]^2
f(50)+f(51)+f(52)+……+f(100)
=50^3+51^3+52^3+100^3+51*(999)
=[100(100+1)/2]²-[49(49+1)/2]²+51(1000-1)
=50*50*101*101-49*49*25*25+51000-51
=(202*202-49*49)*625+51000-51
=(202+49)(202-49)*625+51000-51
=251*153*625+51000-51
=24052824
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