A) \[{{S}_{1}}+{{S}_{3}}={{S}_{2}}\]
B) \[{{S}_{1}}+{{S}_{3}}=2{{S}_{2}}\]
C) \[{{S}_{1}}+{{S}_{2}}=2{{S}_{3}}\]
D) \[{{S}_{1}}+{{S}_{2}}={{S}_{3}}\]
Correct Answer: B
Solution :
We have \[{{a}_{1}}={{a}_{2}}={{a}_{3}}=1\]and \[{{d}_{1}}=1,\ {{d}_{2}}=2,\ {{d}_{3}}=3\]. Therefore, \[{{S}_{1}}=\frac{n}{2}(n+1)\] ......(i) \[{{S}_{2}}=\frac{n}{2}[2n]\] ......(ii) \[{{S}_{3}}=\frac{n}{2}[3n-1]\] ......(iii) Adding (i) and (iii), \[{{S}_{1}}+{{S}_{3}}=\frac{n}{2}[(n+1)+(3n-1)]=2\left[ \frac{n}{2}(2n) \right]=2{{S}_{2}}\] Hence correct relation\[{{S}_{1}}+{{S}_{3}}=2{{S}_{2}}\].
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