Find the second term and $n t h$ term of an AP whose $6 t h$ term is $12$ and $8 t h$ term is $22.$

a_{2}

= - 9,

a_{n}

= 3 n - 43

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a_{2}

= 7,

a_{n}

= 3 n - 52

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a_{2}

= - 8,

a_{n}

= 5 n - 18

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a_{2}

= - 9,

a_{n}

= 5 n - 38

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Solution

The correct option is C $a_{2}$ $= - 8,$ $a_{n}$ $= 5 n - 18$
Let $a$ and $d$ be the first term and the common difference of the given AP respectively. Then,
$a_{6} = 12 \Rightarrow a + 5 d = 12$ ...(1)
$a_{8} = 22 \Rightarrow a + 7 d = 22$ ...(2)
On subtracting (2) from (1), we get
$2 d = 10 \Rightarrow d = 5$
Putting $d = 5$ in (1), we get
$a = 12 - 5 \times 5 = - 13$
Now, $a_{2} = a + d = - 13 + 5 = - 8$ and
$a_{n} = a + (n - 1) d$
$= - 13 + (n - 1) \times 5$
$= 5 n - 18$

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