First slide

Arithmetic progression

Question

If S_n the sum of n terms of A.P, then $S_{n + 3} - 3 S_{n + 2}$ $+ 3 S_{n + 1} - S_{n} =$

Moderate

A

$2 S_{n}$
B

$S_{n + 1}$
C

$3 S_{n}$
D

0

Solution

$\begin{array}{l} S_{n + 3} - 3 S_{n + 2} + 3 S_{n + 1} - S_{n} \\ = (S_{n + 3} - S_{n + 2}) - 2 (S_{n + 2} - S_{n - 1}) + (S_{n + 1} - S_{n}) \\ = T_{n + 3} - 2 T_{n + 2} + T_{n + 1} \\ = (T_{n + 3} - T_{n + 2}) - (T_{n + 2} - T_{n + 1}) \\ = d - d = 0 . \end{array}$

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