If the sum of first n terms of an AP is $3 n$ $^{2}$ $- n$ and its common difference is $6,$ then its first term is

2

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3

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1

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Solution

The correct option is A $2$
Let $S_{n}$ be the sum of $n$ terms of an AP with first term $a$ and common difference $d$ .
Since $S_{n} = 3 n^{2} - n$ and $d = 6$
$\Rightarrow S_{n} = \frac{n}{2} (2 a + (n - 1) d) = 3 n^{2} - n$
$\frac{n}{2} (2 a + (n - 1) 6) = 3 n^{2} - n$
$n (a + (n - 1) 3) = 3 n^{2} - n$
$(a + 3 n - 3) = 3 n - 1$
$a = 2$

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If the sum of first n terms of an AP is 3n2−n and its common difference is 6, then its first term is

The correct option is A 2Let Sn be the sum of n terms of an AP with first term a and common difference d. Since Sn=3n2−n and d=6⇒Sn=n2(2a+(n−1)d)=3n2−n n2(2a+(n−1)6)=3n2−n n(a+(n−1)3)=3n2−n (a+3n−3)=3n−1 a=2

If the sum of first n terms of an AP is $3 n$ $^{2}$ $- n$ and its common difference is $6,$ then its first term is