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 $$\frac{{{l^2} - {a^2}}}{{k - \left( {l + a} \right)}}$$ then k = ?
A. S
B. 2S
C. 3S
D. None of these
Answer: Option B
Solution(By Examveda Team)
$$\eqalign{ & S = \frac{n}{2}\left( {l + a} \right) \cr & l = a + \left( {n - 1} \right)d \cr & d = \frac{{{l^2} - {a^2}}}{{k - \left( {l + a} \right)}}\,{\text{and}}\,{\text{also}}\,d = \frac{{l - a}}{{n - 1}} \cr & \therefore \frac{{l - a}}{{n - 1}} = \frac{{\left( {l + a} \right)\left( {l - a} \right)}}{{k - \left( {l + a} \right)}} \cr & \Rightarrow \frac{1}{{n - 1}} = \frac{{l + a}}{{k - \left( {l + a} \right)}} \cr & \Rightarrow k - \left( {l + a} \right) = \left( {n - 1} \right)\left( {l + a} \right) \cr & \Rightarrow k = \left( {n - 1} \right)\left( {l + a} \right) + \left( {l + a} \right) \cr & \Rightarrow k = \left( {l + a} \right)\left( {n - 1 + 1} \right) \cr & \Rightarrow k = n\left( {l + a} \right) \cr & \, \Rightarrow k = 2 \times \frac{n}{2}\left( {l + a} \right)\left\{ {\therefore \frac{n}{2}\left( {l + a} \right) = S} \right\} \cr & \Rightarrow k = 2 \times S \cr & \Rightarrow k = 2S \cr} $$Related Questions on Progressions
Find the first term of an AP whose 8th and 12th terms are respectively 39 and 59.
A. 5
B. 6
C. 4
D. 3
E. 7
The sum of the first 16 terms of an AP whose first term and third term are 5 and 15 respectively is
A. 600
B. 765
C. 640
D. 680
E. 690
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