Vanuatu
Assume that the symbol ‡x‡ denotes the largest integer not exceeding x. For example, ‡3‡=3, and ‡4.9‡=4.
What is the value of
‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡.
Answer:
38
- Given, ‡x‡ is the largest integer not exceeding x.
Let x be a number which lies between two square numbers a and b, i.e. a<x<b then √x will lie between √a and √b, i.e. √a<√x<√b - Here, square numbers from 1 to 16 are 1,4,9, and 16.
Because, the root of the numbers that are greater than or equal to 1 and less than 4,i.e.1≤x<4 will be greater than or equal to 1 and less than 2,i.e.1≤√x<2
Therefore, ‡√x‡=1 for 1≤x<4
⟹‡√1‡ + ‡√2‡ + ‡√3‡ = 1 + 1 + 1 =3×1=3 - The root of the numbers that are greater than or equal to 4 and less than 9,i.e.2≤x<9 will be greater than or equal to 2 and less than 3,i.e.2≤√x<3
Therefore, ‡√x‡=2 for 4≤x<9
⟹‡√4‡ + ‡√5‡ + ‡√6‡ + ‡√7‡ + ‡√8‡ = 2 + 2 + 2 + 2 + 2 =5×2=10 - Similarly, for 9≤x<16,√x will be 3≤√x<4
Therefore, ‡√x‡=3 for 9≤x<16
⟹‡√9‡ + ‡√10‡ + ‡√11‡ + ‡√12‡ + ‡√13‡ + ‡√14‡ + ‡√15‡ = 3 + 3 + 3 + 3 + 3 + 3 + 3 =7×3=21
And √16=4
⟹‡√16‡=4 - ‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡=3+10+21+4
Hence, the value of ‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡ is 38
