if the sum of p term of an Ap is same as the sum of q term show that the sum of q and p is zero

if the sum of p term of an Ap is same as the sum of q term show that t

1.	if the sum of p term of an Ap is same as the sum of q term show that the sum of q and p is zero
Answer» Sp = Sq{tex} \\Rightarrow \\frac{p}{2}\\left[ {2a + (p - 1)d} \\right] = \\frac{q}{2}\\left[ {2a + (q - 1)d} \\right]{/tex}{tex} \\Rightarrow p\\left[ {2a + pd - d} \\right] = q\\left[ {2a + qd - d} \\right]{/tex}{tex} \\Rightarrow {/tex}\xa02ap + p2d - pd = 2aq + q2d - qd{tex} \\Rightarrow {/tex}\xa02a (p - q) + (p2 - q2)d - (p - q)d = 0{tex} \\Rightarrow {/tex}\xa02a (p - q) + (p + q) (p - q)d - (p - q)d = 0{tex} \\Rightarrow {/tex}\xa0(p - q) [2a + (p + q - 1)d] = 0{tex} \\Rightarrow {/tex}\xa0Sp+q = 0