Question

sa se calculeze:
[rad 1]+ [rad2]+...+[rad100] [ ] inseamna parte intreaga

Answer 1

[√1]+[√2]+...+[√100]=1+1+1+2+2+2+2+2+...+10=3·1+5·2+...+10=∑ k·(2k+1) +10=∑ 2k² +∑ k +10 =2·∑ k² +∑ k +10=2· k(k+1)(2k+1) /6 +k(k+1)/2 +10 =k(k+1)(2k+1) /3 +k(k+1) /2 +10 =2k(k+1)(2k+1) /3 +k(k+1) /2 +10 =k(k+1)(4k+5) /6 +10 dar deoarece k=9 => [√1]+[√2]+...+[√100]=9·10·41 /6 +10= 15·41=10=625 .

Answer 2

{√1]=[√2]=[√3]=1 , 3 numere

[√4]=[√5}=...[√8}=2, 5 numere

[√9}=[√10}=...=[√15]=3, in total7 numere

[√16}=...=[√24]=4, in total 9 numere

[√25}=...=[√35}=5, in total 11 numere

[√36]=...=[√48]=6 in total 13 numere

[√49]=...=[√63]=7, in total 15 numere

[√64]=...=[√80]=8, in total 17 numere

[√81]=...=[√99]=9, in total 19 numere

[√100]=10, un numar

...................................

3*1+5*2+7*3+9*4+11*5+13*6+15*7+17*8+19*9+1*10=

=3+10+21+36+55+78+105+136+171+10=625