If S1,S2,S3 denote the sum of first n1,n2,n3 terms respectively of an A P ,then {S1(n2-n3)}/n1 + {S2(n3-n1)}/n2 +{S3(n1-n2)}/n3 = - Maths - Sequences and Series

Question

If S1,S2,S3 denote the sum of
first n1,n2,n3 terms respectively
of an A P ,then
{S1(n2-n3)}/n1 +
{S2(n3-n1)}/n2
+{S3(n1-n2)}/n3 =
(A) 0 (B) 1 (C) S1S2S3 (D)
n1n2n3 Give the process also

Ajanta Trivedi · Accepted Answer

let the first term of the AP be a and the common difference be
d
sum of n terms of AP is given by : S=n2 2a+(n-1)d
S1=n12 2a+(n1-1)dS1n1=12 2a+(n1-1)d
therefore
the given expression is:
S1(n2-n3)n1+S2(n3-n1)n2+S3(n1-n2)n3=12 2a+(n1-1)d(n2-n3)+12
2a+(n2-1)d(n3-n1)+12 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-n2n1+n3n1-n3n2-(n2-n3+n3-n1+n1-n2=0+d*0=0
thus option (a) is correct

hope this helps you

If S1,S2,S3 denote the sum of first n1,n2,n3 terms respectively of an A.P.,then {S1(n2-n3)}/n1 + {S2(n3-n1)}/n2 +{S3(n1-n2)}/n3 = (A) 0 (B) 1 (C) S1S2S3 (D) n1n2n3 Give the process also .

If S₁,S₂,S₃ denote the sum of first n₁,n₂,n₃ terms respectively of an A.P.,then {S₁(n₂-n₃)}/n₁ + {S₂(n₃-n₁)}/n₂ +{S₃(n₁-n₂)}/n₃ =

(A) 0 (B) 1 (C) S₁S₂S₃ (D) n₁n₂n_{3 Give the process also .}