设等差数列{an}公差d是2,前n项和为Sn,则lim(an^2-n^2)/Sn

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设等差数列{an}公差d是2,前n项和为Sn,则lim(an^2-n^2)/Sn设等差数列{an}公差d是2,前n项和为Sn,则lim(an^2-n^2)/Sn设等差数列{an}公差d是2,前n项和为

设等差数列{an}公差d是2,前n项和为Sn,则lim(an^2-n^2)/Sn
设等差数列{an}公差d是2,前n项和为Sn,则lim(an^2-n^2)/Sn

设等差数列{an}公差d是2,前n项和为Sn,则lim(an^2-n^2)/Sn
S(n)=n[a(1)+a(n)]2=na(1)+n(n-1)d/2=na(1)+n(n-1)
a(n)^2-n^2=[a(1)+(n-1)d]^2-n^2=a(1)^2+4(n-1)a(1)+4(n-1)^2-n^2
所以
lim(a(n)^2-n^2)/S(n)
=lim[na(1)+n(n-1)]/[a(1)^2+4(n-1)a(1)+4(n-1)^2-n^2]
=lim[a(1)/n+(1-1/n)]/{[a(1)/n]^2+4(1-1/n)a(1)/n+4(1-1/n)^2-1}
=(0+1-0)/[0+4×(1-0)×0+4×(1-0)^2-1]
=1/3

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