f(x)=x/3x+1,数列{an}满足a1=1/3,an+1=f(an) (n∈N) (1)求证:数列{1/an}是等差数列,并求an通项公式(2)设sn(x)=x/a1+x的平方/a2+……+x的n次方/an(x>0),求sn(x)
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![f(x)=x/3x+1,数列{an}满足a1=1/3,an+1=f(an) (n∈N) (1)求证:数列{1/an}是等差数列,并求an通项公式(2)设sn(x)=x/a1+x的平方/a2+……+x的n次方/an(x>0),求sn(x)](/uploads/image/z/5630577-33-7.jpg?t=f%28x%29%3Dx%2F3x%2B1%2C%E6%95%B0%E5%88%97%7Ban%7D%E6%BB%A1%E8%B6%B3a1%3D1%2F3%2Can%2B1%3Df%28an%29+%28n%E2%88%88N%EF%BC%89+%EF%BC%881%29%E6%B1%82%E8%AF%81%EF%BC%9A%E6%95%B0%E5%88%97%7B1%2Fan%7D%E6%98%AF%E7%AD%89%E5%B7%AE%E6%95%B0%E5%88%97%2C%E5%B9%B6%E6%B1%82an%E9%80%9A%E9%A1%B9%E5%85%AC%E5%BC%8F%EF%BC%882%EF%BC%89%E8%AE%BEsn%EF%BC%88x%29%3Dx%2Fa1%2Bx%E7%9A%84%E5%B9%B3%E6%96%B9%2Fa2%2B%E2%80%A6%E2%80%A6%2Bx%E7%9A%84n%E6%AC%A1%E6%96%B9%2Fan%28x%3E0%29%2C%E6%B1%82sn%28x%29)
f(x)=x/3x+1,数列{an}满足a1=1/3,an+1=f(an) (n∈N) (1)求证:数列{1/an}是等差数列,并求an通项公式(2)设sn(x)=x/a1+x的平方/a2+……+x的n次方/an(x>0),求sn(x)
f(x)=x/3x+1,数列{an}满足a1=1/3,an+1=f(an) (n∈N) (1)求证:数列{1/an}是等差数列,并求an通项公式
(2)设sn(x)=x/a1+x的平方/a2+……+x的n次方/an(x>0),求sn(x)
f(x)=x/3x+1,数列{an}满足a1=1/3,an+1=f(an) (n∈N) (1)求证:数列{1/an}是等差数列,并求an通项公式(2)设sn(x)=x/a1+x的平方/a2+……+x的n次方/an(x>0),求sn(x)
(1)
证:
a(n+1)=f(an)=an/(3an +1)
1/a(n+1)=(3an +1)/an=1/an +3
1/a(n+1)-1/an=3,为定值.
1/a1=1/(1/3)=3,数列{1/an}是以3为首项,3为公差的等差数列.
1/an =3+3(n-1)=3n
an=1/(3n)
n=1时,a1=1/3,同样满足.
数列{an}的通项公式为an=1/(3n).
(2)
x=1时,Sn(x)=1/a1+1/a2+...+1/an=3(1+2+...+n)=3n(n+1)/2
x≠1时,
Sn=x/a1+x²/a2+...+xⁿ/an=3x+6x²+9x³+...+(3n)xⁿ
xSn=3x²+6x³+...+(3n-3)xⁿ+(3n)x^(n+1)
Sn-xSn=(1-x)Sn=3[x+x²+...+xⁿ-nx^(n+1)]=3[x(1-xⁿ)/(1-x) -nx^(n+1)]
Sn=3x(1-xⁿ)/(1-x)² -3nx^(n+1)/(1-x)