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Assume that the symbol ‡x‡‡x‡ denotes the largest integer not exceeding xx. For example, ‡3‡=3‡3‡=3, and ‡4.9‡=4‡4.9‡=4.
What is the value of
‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡.‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡.
Answer:
3838
- Given, ‡x‡‡x‡ is the largest integer not exceeding xx.
Let xx be a number which lies between two square numbers aa and bb, i.e. a<x<ba<x<b then √x√x will lie between √a√a and √b√b, i.e. √a<√x<√b√a<√x<√b - Here, square numbers from 11 to 1616 are 1,4,9, and 161,4,9, and 16.
Because, the root of the numbers that are greater than or equal to 1 and less than 4,i.e.1≤x<44,i.e.1≤x<4 will be greater than or equal to 1 and less than 2,i.e.1≤√x<22,i.e.1≤√x<2
Therefore, ‡√x‡=1‡√x‡=1 for 1≤x<41≤x<4
⟹⟹‡√1‡‡√1‡ ++ ‡√2‡‡√2‡ ++ ‡√3‡‡√3‡ == 11 ++ 11 ++ 11 =3×1=3=3×1=3 - The root of the numbers that are greater than or equal to 4 and less than 9,i.e.2≤x<99,i.e.2≤x<9 will be greater than or equal to 2 and less than 3,i.e.2≤√x<33,i.e.2≤√x<3
Therefore, ‡√x‡=2‡√x‡=2 for 4≤x<94≤x<9
⟹⟹‡√4‡‡√4‡ ++ ‡√5‡‡√5‡ ++ ‡√6‡‡√6‡ ++ ‡√7‡‡√7‡ ++ ‡√8‡‡√8‡ == 22 ++ 22 ++ 22 ++ 22 ++ 22 =5×2=10=5×2=10 - Similarly, for 9≤x<16,√x9≤x<16,√x will be 3≤√x<43≤√x<4
Therefore, ‡√x‡=3‡√x‡=3 for 9≤x<169≤x<16
⟹⟹‡√9‡‡√9‡ ++ ‡√10‡‡√10‡ ++ ‡√11‡‡√11‡ ++ ‡√12‡‡√12‡ ++ ‡√13‡‡√13‡ ++ ‡√14‡‡√14‡ ++ ‡√15‡‡√15‡ == 33 ++ 33 ++ 33 ++ 33 ++ 33 ++ 33 ++ 33 =7×3=21=7×3=21
And √16=4√16=4
⟹‡√16‡=4⟹‡√16‡=4 - ‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡=3+10+21+4‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡=3+10+21+4
Hence, the value of ‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡‡√1‡+‡√2‡+‡√3‡ +......+‡√16‡ is 3838
