Question

the third term of AP in 3 and 11th term is -21 find its first term and comman difference

Answer 1

$\bold{ \: 3rd\: ter m \: = 3} \\ \\ = > a + 2d = 3 \\ \\ = > a = 3 - 2d \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ...(i) \\ \\ \\ \\ \bold{11th \: term = - 21} \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... (ii) \\ \\ = > a + 10d = - 21 \\ \\ = > a = - 21 - 10d \\ \\ \\ \\ \\ \\ \: \mathbf{both \: (i) \: and \: (ii) \: are \: equal \: to \: a \: } \\ \\ a \: = a \\ \\ = > 3 - 2d = - 21 - 10d \\ \\ = > 10d - 2d = - 21 - 3 \\ \\ = > 8d = - 24 \\ \\ = > d = - 3$

$\mathbf{ Putting \: the \: value \: of \: d \: i n \: ( i ) , }$

$= > a = 3 - 2d \\ \\ = > a= 3 - 2( - 3) \\ \\ = > a= 3 + 6\\ \\ = >a = 9$

$\bold{ \: hence} \\ \\ \bold{ \: common \: difference \: = d \: = - 3 }\\ \\ \bold{first \: term \: = \: a \: = 9}$