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For the product n(n+1)(2n+1),n∈N which one of the following is not necessarily true ?
- a)It is even
- b)Divisible by 3
- c)Divisible by the sum of the squares of first n natural numbers
- d)Never divisible by 237
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
For the productn(n+1)(2n+1),n∈N which one of the following is not...
Clearly the expression is basically the formula for sum of 1st n squares but without the "6" in the denominator.
So options 1,2,4 are true.
And about 237
The above expression will obviously be divisible by 237
When n= 237 as it will be 237*238*475 In fact even for n = 236 and n = 118 it will be divisible by 237.
Most Upvoted Answer
For the productn(n+1)(2n+1),n∈N which one of the following is not...
We can start by expanding the expression:
n(n 1)(2n 1) = 2n^3 + 3n^2 + n
Now, we can use the summation formulas:
∑n = (n(n+1))/2
∑n^2 = (n(n+1)(2n+1))/6
Using these formulas, we can write:
∑n(n 1)(2n 1) = ∑(2n^3 + 3n^2 + n)
= 2∑n^3 + 3∑n^2 + ∑n
= 2[(n(n+1)/2)^2] + 3[(n(n+1)(2n+1))/6] + (n(n+1))/2
= (n^2(n+1)^2)/2 + n(n+1)(2n+1)/2 + n(n+1)/2
= (n(n+1)/2)[n(n+1) + 2(2n+1) + 1]
= (n(n+1)/2)(2n^2 + 5n + 2)
= (n(n+1)(2n+1)(n+2))/3
Therefore, ∑n(n 1)(2n 1) = (n(n+1)(2n+1)(n+2))/3 for any positive integer n.
n(n 1)(2n 1) = 2n^3 + 3n^2 + n
Now, we can use the summation formulas:
∑n = (n(n+1))/2
∑n^2 = (n(n+1)(2n+1))/6
Using these formulas, we can write:
∑n(n 1)(2n 1) = ∑(2n^3 + 3n^2 + n)
= 2∑n^3 + 3∑n^2 + ∑n
= 2[(n(n+1)/2)^2] + 3[(n(n+1)(2n+1))/6] + (n(n+1))/2
= (n^2(n+1)^2)/2 + n(n+1)(2n+1)/2 + n(n+1)/2
= (n(n+1)/2)[n(n+1) + 2(2n+1) + 1]
= (n(n+1)/2)(2n^2 + 5n + 2)
= (n(n+1)(2n+1)(n+2))/3
Therefore, ∑n(n 1)(2n 1) = (n(n+1)(2n+1)(n+2))/3 for any positive integer n.
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For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer?
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For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer?.
For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for For the productn(n+1)(2n+1),n∈N which one of the following is not necessarily true ?a)It is evenb)Divisible by 3c)Divisible by the sum of the squares of first n natural numbersd)Never divisible by 237e)None of theseCorrect answer is option 'D'. Can you explain this answer?.
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