If demotes the sum of first n terms of an A.P., prove that S 12 = 3 (s 8 – S 4 ) .

Question

If demotes the sum of first n terms of an A.P., prove that S 12 = 3 (s 8 &#x2013; S 4 ) .

Philoid · Accepted Answer

S₈ = [2a + 7d]

= 4 (2a + 7d) (i)

S₄ = [2a + 7d]

= 2 (2a + 7d) (ii)

L.H.S = S₁₂ = [2a + 11d]

= 6 [2a + 11d]

R.H.S = 3 (S₈ – S₄)

= 3 [4 (2a + 7d) – 2 (2a + 3d)] [From (i) and (ii)]

= 6 [4a + 14d – 2a – 3d]

= 6 [2a + 11d]

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

Hence, proved