An=25,d=2Sn = 120 so find the nand a .

1.	An=25,d=2Sn = 120 so find the nand a .
Answer» GIVEN :– $\\ \: \: { \huge{.}} \: \: { \bold{ a_{n} = 25}} \\$ $\\ \: \: { \huge{.}} \: \: { \bold{ d=2}} \\$ $\\ \: \: { \huge{.}} \: \: { \bold{ S_{n} = 120}} \\$ TO FIND :– $\\ \: \: { \huge{.}} \: \: { \bold{a=?}} \\$ $\\ \: \: { \huge{.}} \: \: { \bold{n=?}} \\$ SOLUTION :– • We KNOW that – $\\ \dashrightarrow { \bold{ a_{n} = a + (n -1)d}} \\$ • PUT the values – $\\ \implies { \bold{25= a + (n -1)2}} \\$ $\\ \implies { \bold{25= a +2n -2}} \\$ $\\ \implies { \bold{a = 27 - 2n \: \: \: \: - - - - eq.(1)}} \\$ ▪︎ And – $\\ \dashrightarrow { \bold{ S_{n} = \dfrac{n}{2} \{a + a_n \}}} \\$ • So – $\\ \implies { \bold{ 120 = \dfrac{n}{2} \{a +25\}}} \\$ $\\ \implies { \bold{240 =n \{a +25\}}} \\$ • USING eq.(1) – $\\ \implies { \bold{240 =n \{27 - 2n +25\}}} \\$ $\\ \implies { \bold{240 =n (52 - 2n )}} \\$ $\\ \implies { \bold{240 =52n - 2 {n}^{2}}} \\$ $\\ \implies { \bold{2 {n}^{2} - 52n + 240 =0}} \\$ $\\ \implies { \bold{{n}^{2} - 26n + 120 =0}} \\$ $\\ \implies { \bold{{n}^{2} - 20n - 6n+ 120 =0}} \\$ $\\ \implies { \bold{n(n -20) - 6(n - 20) =0}} \\$ $\\ \implies { \bold{(n - 6)(n -20)=0}} \\$ $\\ \implies \large{ \boxed{ \bold{n = 6 \: ,\: 20}}} \\$ • Using eq.(1) – $\\ \implies \large{ \boxed{ \bold{a = 15 \: ,\: - 13}}} \\$

Answer»

GIVEN :–

$\\ \: \: { \huge{.}} \: \: { \bold{ a_{n} = 25}} \\$