A) \[n:(n+1)\]
B) \[(n+1):n\]
C) \[n:(n+2)\]
D) \[(n+2):n\]
Correct Answer: B
Solution :
Let a and d be the first term and common difference respectively of the given A.P. Now, \[{{S}_{1}}\]= Sum of odd terms \[\Rightarrow \] \[{{S}_{1}}={{a}_{1}}+{{a}_{3}}+{{a}_{5}}+....+{{a}_{2n+1}}\] \[\Rightarrow \] \[{{S}_{1}}=\frac{n+1}{2}\left\{ {{a}_{1}}+{{a}_{2n+1}} \right\}\] \[\Rightarrow \] \[{{S}_{2}}=\frac{n}{2}\,\left[ {{a}_{2}}+{{a}_{2n}} \right]\] \[\Rightarrow \] \[{{S}_{2}}=\frac{n}{2}\,\left[ \left( a+d \right)+\left\{ a+\left( 2n-1 \right)d \right\} \right]\] \[\Rightarrow \] \[{{S}_{2}}=n(a+nd)\] \[\therefore \] \[{{S}_{1}}:{{S}_{2}}=(n+1)\,\,(a+nd)\,:\,n(a+nd)\] \[=(n+1):n\]
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