Question

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by \[\frac{l^2 - a^2}{k - (l + a)}\] ,  then k =

Options

• S

• 2S

• 3S

• none of these

Answer 1

2S

Given:

\[S = \frac{n}{2}\left( l + a \right)\]

\[ \Rightarrow \left( l + a \right) = \frac{2S}{n}\]

\[\text{ Also,} d = \frac{l^2 - a^2}{k - \left( l + a \right)}\]

\[ \Rightarrow d = \frac{\left( l + a \right)\left( l - a \right)}{k - \left( l + a \right)}\]

\[ \Rightarrow d = \frac{\left[ \left( n - 1 \right)d \right] \times \frac{2S}{n}}{k - \frac{2S}{n}}\]

\[ \Rightarrow k - \frac{2S}{n} = \left( n - 1 \right)\frac{2S}{n}\]

\[ \Rightarrow k = \frac{2S}{n}\left( n - 1 + 1 \right)\]

\[ \Rightarrow k = 2S\]