Arithmetic progression

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Question

Let S_n denotes the sum of n terms of an A.P. whose first term is a. If the common difference
d = S_n – k S_{n – 1} + S_{n – 2} then k =

Moderate

Solution

We have , $a_{n} = S_{n} - S_{n - 1}$ (1)
and $a_{n - 1} = S_{n - 1} - S_{n - 2}$ (2)
$\begin{array}{l} ∴ d = a_{n} - a_{n - 1} \\ = (S_{n} - S_{n - 1}) - (S_{n - 1} - S_{n - 2}) \\ [From (1) and (2)] \\ = S_{n} - 2 S_{n - 1} + S_{n - 2} . \end{array}$

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