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Question
If the sum of P terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be
Options
0
p − q
p + q
−(p + q)
Solution
In the given problem, we are given Sp = q and Sq = 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 = 2ap + q(q-1)d` ...............(2)
Subtracting (2) from (1), we get
2q - 2p = 2ap + [p ( p - 1) d ] - 2 aq - [q (q-1)d]
2q - 2 p = 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)`
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