If S1 , S2 , S3 are the sum of n terms of three AP's the first term of each - Maths - Arithmetic Progressions

Question

If S1 , S2 , S3 are the sum of
n terms of three AP's the first term of each being unity and
respective common diffrence being 1 , 2 , 3_ _ _ _ _ Prove that
S1+S 3 = 2S2

koabi · Answer

s1= [2a +(n-1)d]n/2
=[2a +n-1)n/2
s2=[2a+(n-1)2]n/2
=[2a+2n-2]n/2 =(2an +2n2 -2n)/2
s3=[2a+(n-1)3]n/2 =(2a+3n-3)n/2
s3 +s1=[2an+3n2 -3n
+2an+n2-n]/2
=(4an+4n2-4n)/2
=2an+2n2 -2=2s2
hence proved

If S1 , S2 , S3 are the sum of n terms of three AP's the first term of each being unity and respective common diffrence being 1 , 2 , 3_ _ _ _ _. Prove that S1+S 3 = 2S2.

If S₁ , S₂ , S₃ are the sum of n terms of three AP's the first term of each being unity and respective common diffrence being 1 , 2 , 3_ _ _ _ _. Prove that S₁+S ₃= 2S2.