1.分解因式:a^4+2a^3b+3a^2b^2+2ab^3+b^4
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1.分解因式:a^4+2a^3b+3a^2b^2+2ab^3+b^4
2.对於任意自然数n,有f(n)=
1/{[三次根号(n^2+2n+10+三次根号(n^2-1)+三次根号(n^-2n+1)]},
则f(1)+f(3)+f(5)+…… +f(999)=?
已知a、b、c、d是正整数,且b/a=(4d-7)/c,(b+1)/a=(7b+7)/c,则c/a的值和d/b的值是多少?
2.对於任意自然数n,有f(n)=
1/{[三次根号(n^2+2n+10+三次根号(n^2-1)+三次根号(n^-2n+1)]},
则f(1)+f(3)+f(5)+…… +f(999)=?
已知a、b、c、d是正整数,且b/a=(4d-7)/c,(b+1)/a=(7b+7)/c,则c/a的值和d/b的值是多少?
1.
a^4+2a^3b+3a^2b^2+2ab^3+b^4
=a^4+b^4+2a^2b^2+2a^3b+a^2b^2+2ab^3
=(a^2+b^2)^2+2ab(a^2+b^2)+a^2b^2
=(a^2+ab+b^2)^2
2.对於任意自然数n,有f(n)=
1/{[三次根号(n^2+2n+10+三次根号(n^2-1)+三次根号(n^-2n+1)]},
则f(1)+f(3)+f(5)+…… +f(999)=?
根据a^3-b^3=(a-b)(a^2+ab+b^2)
(n+1)-(n-1)
=[(n+1)^(1/3)]^3-[(n-1)^(1/3)]^3
=[(n+1)^(1/3)-(n-1)^(1/3)]*{(n+1)^(2/3)+(n^2-1)^(1/3)+(n-1)^(2/3)}
=2
f(n)=1/{(n+1)^(2/3)+(n^2-1)^(1/3)+(n-1)^(2/3)}
=[(n+1)^(1/3)-(n-1)^(1/3)]/2
f(1)=1/4^(1/3)=2^(-2/3)
f(1)+f(3)+f(5)+...+f(999)
=f(1)+(1/2)[4^(1/3)-2^(1/3)+6^(1/3)-4^(1/3)+8^(1/3)-6^(1/3)+...+1000^(1/3)-998^(1/3)]
=f(1)+(1/2)[10-2^(1/3)]
=2^(-2/3)-2^(-2/3)+5
=5
已知a、b、c、d是正整数,且b/a=(4d-7)/c,(b+1)/a=(7b+7)/c,则c/a的值和d/b的值是多少?
b/(b+1)=(4d-7)/7(b+1)
7b=4d-7
d=7(b+1)/4
b/a=(4d-7)/c=7b/c
c/a=7
a^4+2a^3b+3a^2b^2+2ab^3+b^4
=a^4+b^4+2a^2b^2+2a^3b+a^2b^2+2ab^3
=(a^2+b^2)^2+2ab(a^2+b^2)+a^2b^2
=(a^2+ab+b^2)^2
2.对於任意自然数n,有f(n)=
1/{[三次根号(n^2+2n+10+三次根号(n^2-1)+三次根号(n^-2n+1)]},
则f(1)+f(3)+f(5)+…… +f(999)=?
根据a^3-b^3=(a-b)(a^2+ab+b^2)
(n+1)-(n-1)
=[(n+1)^(1/3)]^3-[(n-1)^(1/3)]^3
=[(n+1)^(1/3)-(n-1)^(1/3)]*{(n+1)^(2/3)+(n^2-1)^(1/3)+(n-1)^(2/3)}
=2
f(n)=1/{(n+1)^(2/3)+(n^2-1)^(1/3)+(n-1)^(2/3)}
=[(n+1)^(1/3)-(n-1)^(1/3)]/2
f(1)=1/4^(1/3)=2^(-2/3)
f(1)+f(3)+f(5)+...+f(999)
=f(1)+(1/2)[4^(1/3)-2^(1/3)+6^(1/3)-4^(1/3)+8^(1/3)-6^(1/3)+...+1000^(1/3)-998^(1/3)]
=f(1)+(1/2)[10-2^(1/3)]
=2^(-2/3)-2^(-2/3)+5
=5
已知a、b、c、d是正整数,且b/a=(4d-7)/c,(b+1)/a=(7b+7)/c,则c/a的值和d/b的值是多少?
b/(b+1)=(4d-7)/7(b+1)
7b=4d-7
d=7(b+1)/4
b/a=(4d-7)/c=7b/c
c/a=7
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