Question:
if the sum of first 'p' terms of an AP is 'q' and the sum of first q terms is p . show that sum of first [p+q] terms is -[p+q].
Answer:
Let in an AP, first term is a and common difference is d
Now Sp = q
=> (p/2) *{2a + (p-1)d} = q
=> p*{2a + (p-1)d} = 2q
=> 2ap + p2 d - pd = 2q ................1
Again Sq = p
=> (q/2) *{2a + (q-1)d] = p
=> q*{2a + (q-1)d} = 2p
=> 2aq + q2 d - qd = 2p .........2
subtract equation 1 and 2, we get
2ap + p2 d - pd - (2aq + q2 d - qd) = 2q - 2p
=> 2ap + p2 d - pd - 2aq - q2 d + qd = 2q - 2p
=> 2a(p-q) + (p2 - q2 )d - d (p-q) = -2(p-q)
=> 2a(p-q) + (p - q)*(p+q)*d - d (p-q) = -2(p-q)
=> 2a + (p+q)d - d = -2 (divide by (p-q) on both side)
=> 2a + (p+q -1)d = -2
Now multiply (p+q)/2 on both side, we get
(p+q)/2*{2a + (p+q -1)d} = -2*(p+q)/2
=> Sp+q = -(p+q)
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