If Sn denotes the sum of n terms of an A.P., then Sn + 3 – 3Sn + 2 +

1.	If Sn denotes the sum of n terms of an A.P., then Sn + 3 – 3Sn + 2 + 3Sn + 1 – Sn is equal to(a) 0 (b) 1 (c) 2 (d) 3
Answer» (a) 0 S_{n + 3} – 3S_{n + 2} + 3S_{n + 1} – S_n = (S_{n + 3} – S_{n + 2}) – 2 (S_{n + 2}– S_{n + 1}) + (S_{n + 1} – S_n) = t_{n + 3} – 2t_{n + 2} + t_{n + 1} ... (i) (∵ t_n = S_n – S_{n – 1}) ∵ t_{n + 1}, t_{n + 2}, t_{n + 3} are consecutive terms of an A.P, 2t_{n + 2} = t_{n + 1} + t_{n + 3} ...(ii) ∴ From (i) Reqd. Sum = (t_{n + 3} + t_{n + 1}) – 2t_{n + 2} = 2t_{n + 2} – 2t_{n + 2} = 0 (Using (ii))

If Sn denotes the sum of n terms of an A.P., then Sn + 3 – 3Sn + 2 + 3Sn + 1 – Sn is equal to(a) 0 (b) 1 (c) 2 (d) 3

Answer»

(a) 0

S_{n + 3} – 3S_{n + 2} + 3S_{n + 1} – S_n

= (S_{n + 3} – S_{n + 2}) – 2 (S_{n + 2}– S_{n + 1}) + (S_{n + 1} – S_n)

= t_{n + 3} – 2t_{n + 2} + t_{n + 1} ... (i)

(∵ t_n = S_n – S_{n – 1})

∵ t_{n + 1}, t_{n + 2}, t_{n + 3} are consecutive terms of an A.P,

2t_{n + 2} = t_{n + 1} + t_{n + 3} ...(ii)

∴ From (i)

Reqd. Sum = (t_{n + 3} + t_{n + 1}) – 2t_{n + 2}

= 2t_{n + 2} – 2t_{n + 2} = 0 (Using (ii))