The sum of P terms of an ap is Q and the sum of Q terms is P.Find the sum of( P + Q )terms
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Solution :-
According to the question,
The sum of P terms of an AP is Q and the sum of Q terms is P
As we know that,
Sn = n/2 ( 2a + ( n - 1 )d)
Now, Sum of p terms = q
Sp = p/2 ( 2a + ( p - 1 )d ) = q. eq( 1 )
Sum of Q terms = P
Sq = q/2 ( 2a + ( q - 1 )d) = p. eq( 2)
Now,
Sum of p + q terms
S(p + q) = p + q / 2 * ( 2a + ( p + q - 1 )d) eq( 3 )
From ( 1 ) and ( 2 ) equation we get,
p/q * ( 2a + ( p - 1 )d) = q
= 2a + ( p - 1 )d = 2q/p. eq( 4 )
and,
= q/2 ( 2a + ( q - 1 )d) = p
= 2a + ( q - 1 )d = 2p/q. eq( 5 )
Now,
Subtracting ( 4 ) and ( 5 )
2p/q - 2q/p = ( q - p)d
By taking LCM
= 2 ( p + q ) ( p - q ) / pq = ( q - p)d
= d = -2 ( p + q) / pq
Now, Take eq( 3 )
S(p + q) = p + q /2 ( 2a + ( p + q - 1 )d )
= p + q /2 ( 2a + ( p - 1 )d + qd
Subsitute the value of d
= p + q /2 ( 2q /p + q ( - 2 ( q + p) / qp)
= p + q /2 ( 2q - 2q - 2q / p)
= - ( p + q )
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