Question

If S1, S2, S3 denote respectively the sum of first n1, n2 and n3 terms of an A.P., then
S1/n1(n2 - n3) + S2/n2(n3 - n1) + S3/n3(n1 - n2) is equal to

a) 0
b) n1 + n2 + n3
c)n1*n2*n3
d) S1*S2*S3​

Answer 1

S1, S2, S3 denote respectively the sum of first n1, n2 and n3 terms of an A.P., then ,

S1/n1(n2 - n3) + S2/n2(n3 - n1) + S3/n3(n1 - n2) is equal to

We know the formula for calculating the sum of the terms given by,

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

S1 = n1/2 [ 2a + (n1 - 1) d ]

⇒ S1/n1 = 1/2 [2a + (n1 - 1) d]

S2 = n2/2 [ 2a + (n2 - 1) d ]

⇒ S2/n2 = 1/2 [2a + (n2 - 1) d]

S3 = n3/2 [ 2a + (n3 - 1) d ]

⇒ S3/n3 = 1/2 [2a + (n3 - 1) d]

Now consider,

S1/n1(n2 - n3) + S2/n2(n3 - n1) + S3/n3(n1 - n2)

= 1/2 [2a + (n1 - 1) d] (n2 - n3) + 1/2 [2a + (n2 - 1) d] (n3 - n1) + 1/2 [2a + (n3 - 1) d] (n1 - n2)

= a [ n2 - n3 + n3 - n1 + n1 - n2 ] + d  [ (n1 - 1) (n2 - n3)  + (n2 - 1) (n3 - n1) + (n3 - 1) (n1 - n2) ]

= a [ 0 ] + d [ n1n2 - n1n3  + n2n3 - n1n2 + n3n1 - n3n2 - n1 + n2 - n3 + n1 + n2 - n2 ]

= 0 + d [ 0 ]

= 0

∴ S1/n1(n2 - n3) + S2/n2(n3 - n1) + S3/n3(n1 - n2) = 0

Option a is correct.