Statement-I: If the sum of n terms of a seque | vedclass

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(D) For a sequence to be an $AP$,the sum of $n$ terms $S_n$ must be of the form $An^2 + Bn$.In Statement-$I$,$S_n = 6n^2 + 3n + 1$. Since there is a c

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Statement-$I$ is false. Statement-$II$ is true.

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(D) For a sequence to be an $AP$,the sum of $n$ terms $S_n$ must be of the form $An^2 + Bn$.In Statement-$I$,$S_n = 6n^2 + 3n + 1$. Since there is a constant term $(1)$,this cannot be an $AP$. Thus,Statement-$I$ is false.In Statement-$II$,the sum of $n$ terms of an $AP$ with first term $a$ and common difference $d$ is given by $S_n = \frac{n}{2}[2a + (n-1)d] = \frac{d}{2}n^2 + (a - \frac{d}{2})n$. This is clearly of the form $an^2 + bn$. Thus,Statement-$II$ is true.Therefore,Statement-$I$ is false and Statement-$II$ is true.

Statement-$I$: If the sum of $n$ terms of a sequence is $6n^2 + 3n + 1$,then it is an Arithmetic Progression $(AP)$.
Statement-$II$: The sum of $n$ terms of an Arithmetic Progression is always in the form $an^2 + bn$.

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Statement-$I$: If the sum of $n$ terms of a sequence is $6n^2 + 3n + 1$,then it is an Arithmetic Progression $(AP)$.Statement-$II$: The sum of $n$ terms of an Arithmetic Progression is always in the form $an^2 + bn$.