Question

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

Answer 1

Given sn denotes the sum of first n terms of an AP.

Let a be the first term and d be the common difference of the given AP.

Then sn = n/2(2a + (n-1)d).

 

LHS :

S30 = (30/2)(2a + (30 - 1)d)

       = 15(2a + 29d)

       = 30a + 435d.   ----------------  (1).

RHS:

(S20 - S10)  = (20/2)(2a + (20 - 1) * d) - (10/2)(2a + (10 - 1) * d)

                    = 10(2a + 19d) - 5(2a + 9d)

                    = 20a + 190d - 10a - 45d

                    = 10a + 145d      

3(S20 - S10) = 3(10a + 145d)

                    = 30a + 435d   ------------- (2)

Therefore From (1) and (2),  It is proved that S30 = 3(S20 - S10).

LHS = RHS.

Hope this helps!