Question

设an表示与根号n(n∈N*)最接近整数,则1/a1+1/a2+1/a3+.+1/a2006的值为

Answer 1

(k+1/2)^2=k^2+k+1/4,(k-1/2)^2=k^2-k+1/4 ∴a(k^2-k+1)=a(k^2-k+2)=a(k^2-k+3)=……=a(k^2+k)=k,共(k^2+k)-(k^2-k+1)+1=2k个 ∴1/a(k^2-k+1)+1/a(k^2-k+2)+1/(k^2-k+3)+……+1/a(k^2+k)=2k/k=2 即1/a1+1/a2=2,1/a3+1/a4+1/a5+1/a6=2,…… ,1/a1893+1/a1894+……+1/a1980=2(k=44) ∴1/a1+1/a2+1/a3+.+1/a2006=2*44+1/a1981+1/a1982+……+1/a2006 =88+(2006-1981+1)/45=88+26/45=3986/45