Question

一道初中数学题

设S1=1+1/1²+1/2²，S2=1+1/2²+1/3²，S3=1+1/3²+1/4²，……Sn=1+1/n²+1/（n+1）²
设S=根号S1+根号S2+……+根号Sn 则S= （用含n的代数式表示，其中n为正整数）

Answer 1

解：Sn=1+1/n²+1/（n+1）²
={n²（n+1）²+n²+（n+1）²}/{n²（n+1）²}
={n²（n+1）²+2n(n+1)+1}/{n²（n+1）²}
={n(n+1)+1}²/{n²（n+1）²}
所以S=根号S1+根号S2+……+根号Sn
=3/2+7/6+........+{n(n+1)+1}/{n(n+1)}
={1+1/(1*2)}+{1+1/(2*3)}+.......{1+1/(n(n+1))}
=n+1/(1*2)+1/(2*3)+.....1/{n(n+1)}
=n+(1-1/2)+(1/2-1/3)+.....+(1/n-1/(n+1))
=n+1-1/2+1/2+1/3-1/4+1/4+....+1/n-1/(n+1)
=n+1-1/(n+1)
=n(n+2)/(n+1)

Answer 2

S=（6+根号21）|4+根号（n4次方+2n3次方+2n平方+2n+1）|n（n+1)

Answer 3

S=（6+根号21）|4+根号（n4次方+2n3次方+2n平方+2n+1）|n（n+1)

Answer 4

见图片

Answer 5

好难呀