If demotes the sum of the first n terms of an A.P., prove that S 30 = 3 (S 20 – S 10 ) .

Question

If demotes the sum of the first n terms of an A.P., prove that S 30 = 3 (S 20 &#x2013; S 10 ) .

Philoid · Accepted Answer

Proof: S₂₀ = [2a + 19d]

= 10 [2a + 19d] (i)

S₁₀ = [2a + 9d]

= 5 [2a + 9d] (ii)

L.H.S: S₃₀ = [2a + 29d]

= 15 [2a + 29d]

R.H.S: 3 (S₂₀ – S₁₀)

= 3 [10 (2a + 19d) – 5 (2a + 9d)] {From (i) and (ii)}

= 15 [4a + 38d – 2a – 9d]

= 15 [2a + 29d]

Since, L.H.S = R.H.S

Hence, proved