【高中数学/等差数列/数列求和】(2025.全国I卷)设数列{an}满足a1=3,an+1/n=an/(n+1)+1/n/(n+1) (1)证明凯{n*an}为等差数列; (2)给定正整数m,设f(x)=a1*x+a2*x^2+...+am*x^m,求f'(-2).

【问题】

(2025.全国I卷)设数列{an}满足a1=3,an+1/n=an/(n+1)+1/n/(n+1)
(1)证明凯{n*an}为等差数列;
(2)给定正整数m,设f(x)=a1*x+a2*x^2+...+am*x^m,求f'(-2).

【答案】

(1){n*an}的公差为1;
(2)f'(-2)=7/9-1/9*(3m+7)*(-2)^m

【解答】

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