Question

If Sn denotes the sum of the first n terms of an AP, prove that S30 = 3(S20 - S10)

Answer 1

Before all please realize youve typed the question wrong.. in which i wasted my time... later on i realized the question is S30 = 3(S20- S10) and youve written 210 instead or S10.. anyways its fine :'DDD

Sn=n/2(2a+(n-1)d
S30= 30/2(2a+29d) 
      = 30a+435d       ----------------------------1

Similarly,

S20-S10 = 10/2(2a+19d) - 10/2(2a+9d) 
              = 20a+ 190d - 10a - 45d
              = 10a + 145d  --------------------2
equation2 x 3 = 30a+435d --------------------3
​therefore from 1,2 and 3 we get ,

s30 = 3(S20 - S10)