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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
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Solution
2S
Given:
\[S = \frac{n}{2}\left( l + a \right)\]
\[ \Rightarrow \left( l + a \right) = \frac{2S}{n}\]
\[\text{ Also,} d = \frac{l^2 - a^2}{k - \left( l + a \right)}\]
\[ \Rightarrow d = \frac{\left( l + a \right)\left( l - a \right)}{k - \left( l + a \right)}\]
\[ \Rightarrow d = \frac{\left[ \left( n - 1 \right)d \right] \times \frac{2S}{n}}{k - \frac{2S}{n}}\]
\[ \Rightarrow k - \frac{2S}{n} = \left( n - 1 \right)\frac{2S}{n}\]
\[ \Rightarrow k = \frac{2S}{n}\left( n - 1 + 1 \right)\]
\[ \Rightarrow k = 2S\]
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