JEE Exam > JEE Questions > In an A.P., sum of first p terms is q and sum...
Start Learning for Free
In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms is
- a)0
- b)- ( p + q)
- c)p + q
- d)none of these
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
In an A.P., sum of first p terms is q and sum of first q terms is p. S...
Let the first term of the given AP be ‘a' and the common difference be ‘d'. Then, the sum of first ‘n' terms of the AP is given by:
Sn = n/2 {2a+(n-1)d} …….(1)
Here, it is given that:
Sp = q and Sq = p
Using (1), we get:-
q = p/2 {2a+(p-1)d}
and p = q/2 {2a+(q-1)d}
i.e. 2a + (p-1)d = 2q/p …..(2)
and 2a + (q-1)d = 2p/q …..(3)
Subtracting (3) from (2), we get:
(p - 1 - q + 1)d = 2q/p - 2p/q
So, d = 2(q2 - p2)/pq(p-q)
i.e. d = -2(p+q)/pq
Now, substituting the value of ‘d' in eq.n (2), we get:
2a + (p-1){-2(p+q)/pq} = 2q/p
i.e. 2a= 2q/p + 2(p-1)(p+q)/pq
This gives:
a = (p2 + q2 - p - q + pq)/pq
So, we have
Sp+q = (p+q)/2 { 2(p2+q2-p-q+pq)/pq - (p+q-1) 2 (p+q)/pq}
i.e. Sp+q = (p+q)/pq { p2+q2-p-q+pq-p2-pq-qp-q2+p+q}
So, Sp+q = -(p+q)
Sn = n/2 {2a+(n-1)d} …….(1)
Here, it is given that:
Sp = q and Sq = p
Using (1), we get:-
q = p/2 {2a+(p-1)d}
and p = q/2 {2a+(q-1)d}
i.e. 2a + (p-1)d = 2q/p …..(2)
and 2a + (q-1)d = 2p/q …..(3)
Subtracting (3) from (2), we get:
(p - 1 - q + 1)d = 2q/p - 2p/q
So, d = 2(q2 - p2)/pq(p-q)
i.e. d = -2(p+q)/pq
Now, substituting the value of ‘d' in eq.n (2), we get:
2a + (p-1){-2(p+q)/pq} = 2q/p
i.e. 2a= 2q/p + 2(p-1)(p+q)/pq
This gives:
a = (p2 + q2 - p - q + pq)/pq
So, we have
Sp+q = (p+q)/2 { 2(p2+q2-p-q+pq)/pq - (p+q-1) 2 (p+q)/pq}
i.e. Sp+q = (p+q)/pq { p2+q2-p-q+pq-p2-pq-qp-q2+p+q}
So, Sp+q = -(p+q)
Free Test
FREE
| Start Free Test |
Community Answer
In an A.P., sum of first p terms is q and sum of first q terms is p. S...
Explanation:
First, let's denote the first term of the A.P. as 'a' and the common difference as 'd'.
Sum of first p terms:
The sum of the first p terms is given by:
\[ \frac{p}{2} [2a + (p-1)d] = q \]
Sum of first q terms:
The sum of the first q terms is given by:
\[ \frac{q}{2} [2a + (q-1)d] = p \]
Now, we need to find the sum of the first p + q terms:
Sum of first p + q terms:
The sum of the first p + q terms is given by:
\[ \frac{p+q}{2} [2a + (p+q-1)d] \]
Using the above two equations, we can substitute the values of q and p to simplify the expression for the sum of the first p + q terms.
After simplifying, we get:
\[ \frac{p+q}{2} [2a + (p+q-1)d] = - (p + q) \]
Therefore, the sum of the first p + q terms is -(p + q), which corresponds to option 'B'.
First, let's denote the first term of the A.P. as 'a' and the common difference as 'd'.
Sum of first p terms:
The sum of the first p terms is given by:
\[ \frac{p}{2} [2a + (p-1)d] = q \]
Sum of first q terms:
The sum of the first q terms is given by:
\[ \frac{q}{2} [2a + (q-1)d] = p \]
Now, we need to find the sum of the first p + q terms:
Sum of first p + q terms:
The sum of the first p + q terms is given by:
\[ \frac{p+q}{2} [2a + (p+q-1)d] \]
Using the above two equations, we can substitute the values of q and p to simplify the expression for the sum of the first p + q terms.
After simplifying, we get:
\[ \frac{p+q}{2} [2a + (p+q-1)d] = - (p + q) \]
Therefore, the sum of the first p + q terms is -(p + q), which corresponds to option 'B'.
|
Explore Courses for JEE exam
|
|
Top Courses for JEEView all
Question Description
In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer?.
In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer?.
Solutions for In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. 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 In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer?, a detailed solution for In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? has been provided alongside types of In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice In an A.P., sum of first p terms is q and sum of first q terms is p. Sum of its p + q terms isa)0b)- ( p + q)c)p + qd)none of theseCorrect answer is option 'B'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
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