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‡.


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Answer:

3838

Step by Step Explanation:
  1. 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 √xx will lie between √aa and √bb, i.e. √a<√x<√ba<x<b
  2. 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
  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
  4. 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=416=4
    ⟹‡√16‡=4⟹‡16‡=4
  5. ‡√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
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