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Question
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
Options
1
2
3
none of these
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Solution
2
Let the A.P. be a, a+d, a+2d, a+3d...
Given:
\[d = S_n - k S_{n - 1} + S_{n - 2}\]
For n = 3, we have:
\[d = \left( 3a + 3d \right) - k\left( 2a + d \right) + a\]
\[ \Rightarrow 4a + 2d - k\left( 2a + d \right) = 0\]
\[ \Rightarrow 2\left( 2a + d \right) = k\left( 2a + d \right)\]
\[ \Rightarrow 2 = k\]
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