The third term of an A.P is $- 11$ and the nineth term is $- 35$ , find the $n^{t h}$ term of the A.P.

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Solution

$T_{a} + (n - 1) d$
$T_{3} = a + (3 - 1) d$
$T_{3} = a + 2 d = - 11$ ...(1)
Similarly
$T_{9} = a + 8 d = - 35$ ...(2)
(2)-(1)
$(a + 8 d) - (a + 2 d) = - 35 - (- 11)$
$a + 8 d - a - 2 d = - 24$
$6 d = - 24$
$d = - 4$
From (1)
$a + 2 d = - 11$
$a + 2 (- 4) = - 11$
$a = - 11 + 8$
$a = - 3$
$T_{n} = a + (n - 1) d$
$= - 3 + (n - 1) (- 4)$
$= - 3 - 4 n + 4$

$∴ T_{n} = 1 - 4 n$

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