For x∈R, let [x] denote the greatest integer ≤x, then the sum of the series [−13]+[−13−1100]+[−13−2100]+....+[−13−99100] is :
For x∈R, let [x] denote the greatest integer ≤x, then the sum of the series [−13]+[−13−1100]+[−13−2100]+....+[−13−99100] is :
Given:
[−13]+[−13−1100]+[−13−2100]+....+[−13−99100]
where x∈R,[x] denote the greatest integer ≤x,
Since [−13]=−1 and
−13−x100=−100+3x300
therefore,
For case 1-
100+3x<300
⇒x<66.67
∴[−13−x100]=−1
For case 2
300≤100+3x<600
⇒2003≤x<5003
⇒67≤x<166
∴[−13−x100]=−2
So the sum of the series equals
=(−1−1−1−⋯67 times)+(−2−2−2−⋯33 times)
−67−2(33)=−133
(OR)
[−13]+[−13−1100]+[−13−2100]+....+[−13−99100]
where x∈R,[x] denote the greatest integer ≤x,
Here,
[−13]+[−13−1100]+[−13−2100]+....+[−13−66100]
The of G.I.F in all above terms reach to −1.
i.e.,
=−1 −1 −1−...67 times
=−1×67
=−67---(1)
Now, lets take,
[−13−67100]+[−13−68100]+....+[−13−99100]
The of G.I.F in all above terms reach to −2.
=−2−2−2−....33 times.
=−2×33
=−66 ---(2)
⇒[−13]+[−13−1100]+[−13−2100]+....+[−13−99100]=−67−66=−133
Hence, option B is correct.
Given:
[−13]+[−13−1100]+[−13−2100]+....+[−13−99100]
where x∈R,[x] denote the greatest integer ≤x,
Since [−13]=−1 and
−13−x100=−100+3x300
therefore,
For case 1-
100+3x<300
⇒x<66.67
∴[−13−x100]=−1
For case 2
300≤100+3x<600
⇒2003≤x<5003
⇒67≤x<166
∴[−13−x100]=−2
So the sum of the series equals
=(−1−1−1−⋯67 times)+(−2−2−2−⋯33 times)
−67−2(33)=−133
(OR)
[−13]+[−13−1100]+[−13−2100]+....+[−13−99100]
where x∈R,[x] denote the greatest integer ≤x,
Here,
[−13]+[−13−1100]+[−13−2100]+....+[−13−66100]
The of G.I.F in all above terms reach to −1.
i.e.,
=−1 −1 −1−...67 times
=−1×67
=−67---(1)
Now, lets take,
[−13−67100]+[−13−68100]+....+[−13−99100]
The of G.I.F in all above terms reach to −2.
=−2−2−2−....33 times.
=−2×33
=−66 ---(2)
⇒[−13]+[−13−1100]+[−13−2100]+....+[−13−99100]=−67−66=−133
Hence, option B is correct.
