JEE Exam > JEE Questions > Let Vr denote the sum of first r terms of an ...
Start Learning for Free
Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....
Tr is always
Tr is always
- a)an odd number
- b)an even number
- c)a prime number
- d)a composite number
Correct answer is option 'D'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Let Vr denote the sum of first r terms of an arithmetic progression (A...
Most Upvoted Answer
Let Vr denote the sum of first r terms of an arithmetic progression (A...
We have the first term of the A.P. as r and the common difference as (2r - 1).
Thus, the second term would be:
r + (2r - 1) = 3r - 1
The third term would be:
r + 2(2r - 1) = 5r - 3
The fourth term would be:
r + 3(2r - 1) = 7r - 5
And so on.
Thus, the sum of the first r terms of this A.P. would be:
Vr = r + (3r - 1) + (5r - 3) + (7r - 5) + ... + (2r(r-1) - (2r - 3))
We can simplify this expression by grouping the terms as follows:
Vr = (r + (3r - 1) + (5r - 3) + (7r - 5) + ... + (2r(r-1) - (2r - 3))) + (-1 - 2 - 3 - 4 - ... - (r - 2))
The first part of this expression is simply the sum of the first r terms of an A.P. with first term r and common difference (2r - 1). We can use the formula for the sum of an A.P. to simplify this part:
Vr = (r/2)(2r + (r - 1)(2r - 1)) + (-1 - 2 - 3 - 4 - ... - (r - 2))
Simplifying the first part further, we get:
Vr = r(3r - 1) + (-1 - 2 - 3 - 4 - ... - (r - 2))
Now, we can use the formula for the sum of the first n natural numbers to simplify the second part:
Vr = r(3r - 1) - [(r - 2)(r - 1)/2]
Expanding and simplifying the second part, we get:
Vr = r(3r - 1) - (r^2 - 3r + 2)/2
Vr = (7r^2 - 7r + 2)/2
Thus, the sum of the first r terms of the given A.P. is (7r^2 - 7r + 2)/2.
Thus, the second term would be:
r + (2r - 1) = 3r - 1
The third term would be:
r + 2(2r - 1) = 5r - 3
The fourth term would be:
r + 3(2r - 1) = 7r - 5
And so on.
Thus, the sum of the first r terms of this A.P. would be:
Vr = r + (3r - 1) + (5r - 3) + (7r - 5) + ... + (2r(r-1) - (2r - 3))
We can simplify this expression by grouping the terms as follows:
Vr = (r + (3r - 1) + (5r - 3) + (7r - 5) + ... + (2r(r-1) - (2r - 3))) + (-1 - 2 - 3 - 4 - ... - (r - 2))
The first part of this expression is simply the sum of the first r terms of an A.P. with first term r and common difference (2r - 1). We can use the formula for the sum of an A.P. to simplify this part:
Vr = (r/2)(2r + (r - 1)(2r - 1)) + (-1 - 2 - 3 - 4 - ... - (r - 2))
Simplifying the first part further, we get:
Vr = r(3r - 1) + (-1 - 2 - 3 - 4 - ... - (r - 2))
Now, we can use the formula for the sum of the first n natural numbers to simplify the second part:
Vr = r(3r - 1) - [(r - 2)(r - 1)/2]
Expanding and simplifying the second part, we get:
Vr = r(3r - 1) - (r^2 - 3r + 2)/2
Vr = (7r^2 - 7r + 2)/2
Thus, the sum of the first r terms of the given A.P. is (7r^2 - 7r + 2)/2.
Free Test
FREE
| Start Free Test |
Community Answer
Let Vr denote the sum of first r terms of an arithmetic progression (A...
Correct answer is option.D)a composite number
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer?
Question Description
Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer?.
Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let Vr denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r – 1). Let Tr = V r + 1 – Vr – 2 and Qr = Tr + 1 – Tr for r = 1, 2,.....Tr is alwaysa)an odd numberb)an even numberc)a prime numberd)a composite numberCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup