A) \[\frac{{{l}^{2}}-{{a}^{2}}}{2S-(l+a)}\]
B) \[\frac{{{l}^{2}}-{{a}^{2}}}{2S-(l-a)}\]
C) \[\frac{{{l}^{2}}+{{a}^{2}}}{2S+(l+a)}\]
D) \[\frac{{{l}^{2}}+{{a}^{2}}}{2S-(l+a)}\]
Correct Answer: A
Solution :
Key Idea: If a be the first term and \[l\]be the last term of an AP, then \[S=\frac{n}{2}(a+l).\] Let d be the common difference of an AP, then \[S=\frac{n}{2}(a+l)\] ?(i) Also \[l=a+(n-1)\,d\] \[\Rightarrow \] \[d=\frac{l-a}{n-1}\] \[\Rightarrow \] \[d=\frac{l-a}{\frac{2S}{a+l}-1}\] [from (i)] \[=\frac{(l-a)(a+l)}{2S-a-l}\] \[=\frac{{{l}^{2}}-{{a}^{2}}}{2S-a-l}\]
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