Question

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

Options

• 1

• 2

• 3

• none of these

Answer 1

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:

\[d = \left( 3a + 3d \right) - k\left( 2a + d \right) + a\]

\[ \Rightarrow 4a + 2d - k\left( 2a + d \right) = 0\]

\[ \Rightarrow 2\left( 2a + d \right) = k\left( 2a + d \right)\]

\[ \Rightarrow 2 = k\]