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In an Ap. Sp = Q, Sq = P and Sr Denotes the Sum of First R Terms. Then, Sp+Q is Equal to - Mathematics

MCQ

In an AP. Sp = q, Sq = p and Sr denotes the sum of first r terms. Then, Sp+q is equal to

Options

  • 0

  • −(p + q)

  •  p + q

  • pq

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Solution

In the given problem, we are given `S_p - q` and  `S_q = p`

We need to find  `S_( p +q)`

Now, as we know,

`S_n = n/2 [ 2a + (n-1) d]`

So,

`S_p = p/2 [2a + (p-1) d]`

`q = p/2 [2a + (p-1)d]`

2q = 2ap + p (p - 1)d                        .................(1) 

Similarly,

`S_q = q/2 [2a + (q - 1) d ]`

`p = q/2 [ 2a + (q - 1) d ]`

2p = 2aq  + q(q-1)d                          ......................(2)

Subtracting (2) from (1), we get

2q - 2p = 2ap + [p(p - 1)d ] - 2aq - [q(q - 1 ) d ]

2q - 2p = 2a ( p - q) + [ p(p - 1 ) - q( q - 1 ) d

-2(p-q) = 2a(p-q) + [(p2 - q2)-(p-q)] 

       - 2 = 2a + ( p + q - 1 ) d                           ..................(3) 

Now,

`S_(p + q) = ( p+q)/2 [2a + ( p+q - 1 ) d ]`

`S_(p+q) = ((p+q))/2 (-2)   `             ....(Using 3 ) 

`S_(p+q) = - (p+ q) `

Thus,  `S_(p+q) = - (p+ q) `

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Class 10 Maths
Chapter 5 Arithmetic Progression
Q 16 | Page 58
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