Question

please help me out to find it ​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{p_{th}\:term=6P-2}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline\bold{Given :}} \\ \tt: \implies s_{n} = 3 {n}^{2} + n \\ \\ \red{\underline\bold{To \: Find :}} \\ \tt: \implies p_{th} \: term \: of \: A.P = ?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies s_{n} = {3n}^{2} + n \\ \\ \circ \tt{\: where \: n = 1,2,3,4,....} \\ \\ \bold{For \: s_{1} : } \\ \tt: \implies s_{1} = {3 \times 1}^{2} + 1 \\ \\ \tt: \implies s_{1} = 4 \\ \\ \tt: \implies s_{1} = a_{1} = 4\\ \\ \bold{For \: s_{2} : } \\ \\ \tt : \implies s_{2} = {3 \times 2}^{2} + 2 \\ \\ \tt : \implies s_{2} = 14 \\ \\ \tt : \implies a_{2} = s_{2} - s_{1} \\ \\ \tt : \implies a_{2} =14 - 4 \\ \\ \green{\tt : \implies a_{2} =10}$

$\\ \bold{For \: s_{3} : } \\ \tt : \implies s_{3} =3 \times {3}^{2} + 3 \\ \\ \tt : \implies s_{3} =30 \\ \\ \tt : \implies a_{3} = s_{3} - s_{2} \\ \\ \tt : \implies a_{3} =30 - 14 \\ \\ \green{\tt : \implies a_{3} =16} \\ \\ \green{\tt: {\therefore A.P = 4,10,16,22,...}} \\ \\ \circ \: \tt{Common \: difference = 6} \\ \\ \bold{For \: p_{th} \: term : } \\ \tt: \implies p_{th} = a + (P - 1)d \\ \\ \tt: \implies p_{th} =4 + (P - 1) \times 6 \\ \\ \tt: \implies p_{th} =4 + 6P - 6 \\ \\ \green{\tt: \implies p_{th} =6P - 2}$