` a_n =4, d= 2` और ` S_n =-14` दिया है| n और a ज्ञात कीजिए| एक `A.P.` में,

Explanation

` " " a_n =4,d=2 ` और ` " " S_n =-14`
`" " a_n =4" " rArr " " a+ (n-1)d=4 `
` rArr " " a+ (n-1) *2= 4 `
` " " a+ 2n - 2 =4`

` rArr " " a+2n =6 `
तथा ` " " S_n =-14`
` rArr " " (n)/(2) (a+a_n) =-14 `
` rArr " "(n)/(2) (a+4) =-14`
` rArr " " (n)/(2) (6-2n +4)=-14`
` rArr" " (n)/(2) (6-2n + 4) =-14 " " ("समी०" )(1) ("से ")`
` rArr" " (n)/(2) (10-2n) =-14`
` rArr " " n( 5-n ) =-14 `
` rArr " " 5n -n^(2) =-14`

` rArr " " 0 =n^(2) -5n -14`
` rArr " " n^(2) -7n +2n -14 =0`
` rArr " " n(n-7) +2 (n-7) =0`
`rArr " " (n-7) (n+2) =0 `
` rArr " " n -7 =0` या ` n+ 2=0`
` rArr " " n= 7` या `" " n=-2 `
` " " n=-2` सम्भव नहीं है|
` therefore " " n = 7`
समीकरण (1 ) से
` " " a+ 2xx 7=6 `
` rArr " " a= 6-14 =-8`
` therefore " "n= 7 ` तथा a =-8