CAT Exam > CAT Questions > If[log10 1] + [log10 2] + [log10 4] +...+ [lo...
Start Learning for Free
If [log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then
(2016)
- a)96 ≤ n < 104
- b)104 < - n < 107
- c)107 ≤ n < 111
- d)111 ≤ n < 116
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] deno...
[log10 x] = 0, for any value of x ∈ {1, 2, ......9), ...(1)
Similarly [log10 x] = 1, for x ∈ {10, 11, 12 ... 99}...(2)
and [log10x] = 2, for
x ∈ {100, 101, 102, ... 999} ...(3)
Now consider, 1 ≤ n ≤ 9, then
[log101] + [log102] + [log103] ... [log10 n] = 0 (i.e., ≠ n)
Hence the expression given in the question cannot be satisfied.
Now consider, 10 ≤ n ≤ 99, then [log101] + [log102] ... [log10n]
From (1) and (2), the above expression becomes (0 + 0 ... 9 times) + (1 + 1 + ... (n – 9) times) = n – 9
Using the same approach, for
100 ≤ n ≤ 999, [log 101] + [log102] ... [log 10n] = 90 + 2(n – 99)
If can be seen that, only for the third case i.e., 100 ≤ n ≤ 999, can the expression given in the question be satisfied.
Hence 90 + 2(n – 99) = n
⇒ n = 198 – 90 = 108
∴ 107 ≤ n < 111.
Similarly [log10 x] = 1, for x ∈ {10, 11, 12 ... 99}...(2)
and [log10x] = 2, for
x ∈ {100, 101, 102, ... 999} ...(3)
Now consider, 1 ≤ n ≤ 9, then
[log101] + [log102] + [log103] ... [log10 n] = 0 (i.e., ≠ n)
Hence the expression given in the question cannot be satisfied.
Now consider, 10 ≤ n ≤ 99, then [log101] + [log102] ... [log10n]
From (1) and (2), the above expression becomes (0 + 0 ... 9 times) + (1 + 1 + ... (n – 9) times) = n – 9
Using the same approach, for
100 ≤ n ≤ 999, [log 101] + [log102] ... [log 10n] = 90 + 2(n – 99)
If can be seen that, only for the third case i.e., 100 ≤ n ≤ 999, can the expression given in the question be satisfied.
Hence 90 + 2(n – 99) = n
⇒ n = 198 – 90 = 108
∴ 107 ≤ n < 111.
Community Answer
If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] deno...
Notice that $n$ must be greater than or equal to $2$ since $[log10 1]$ equals $0.$
If $2\leq n < 10,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="1,$" which="" is="" less="" than="" />
If $10\leq n < 100,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="" +="" 2="3,$" which="" is="" less="" than="" />
If $100\leq n < 1000,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="" +="" 2="" +="" 3="6,$" which="" is="" less="" than="" />
If $1000\leq n < 10000,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="" +="" 2="" +="" 3="" +="" 4="10,$" which="" is="" less="" than="" />
Since $2016$ is between $1000$ and $10000,$ we know that $[log10 1] + [log10 2] + \cdots + [log10 2016] < 2016.$="" therefore,="" the="" equation="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="n$" is="" never="" satisfied="" for="" $n="" />
Thus, the answer is $\boxed{\text{none of the above}}.$
If $2\leq n < 10,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="1,$" which="" is="" less="" than="" />
If $10\leq n < 100,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="" +="" 2="3,$" which="" is="" less="" than="" />
If $100\leq n < 1000,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="" +="" 2="" +="" 3="6,$" which="" is="" less="" than="" />
If $1000\leq n < 10000,$="" then="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="" \leq="" 0="" +="" 0="" +="" \cdots="" +="" 0="" +="" 1="" +="" 2="" +="" 3="" +="" 4="10,$" which="" is="" less="" than="" />
Since $2016$ is between $1000$ and $10000,$ we know that $[log10 1] + [log10 2] + \cdots + [log10 2016] < 2016.$="" therefore,="" the="" equation="" $[log10="" 1]="" +="" [log10="" 2]="" +="" \cdots="" +="" [log10="" n]="n$" is="" never="" satisfied="" for="" $n="" />
Thus, the answer is $\boxed{\text{none of the above}}.$
|
Explore Courses for CAT exam
|
|
Similar CAT Doubts
Top Courses for CATView all
If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer?
Question Description
If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer?.
If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer?.
Solutions for If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer?, a detailed solution for If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? has been provided alongside types of If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice If[log10 1] + [log10 2] + [log10 4] +...+ [log10 n] = n where [x] denotes the greatest integer less than or equal to x, then(2016)a)96 ≤ n < 104b)104 < - n < 107c)107 ≤ n < 111d)111 ≤ n < 116Correct answer is option 'C'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Signup to solve all Doubts
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup