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Question
Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.
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Solution
Given:
\[a_6 = 12\]
\[ \Rightarrow a + \left( 6 - 1 \right)d = 12\]
\[ \Rightarrow a + 5d = 12 . . . (i)\]
\[ a_8 = 22\]
\[ \Rightarrow a + \left( 8 - 1 \right)d = 22\]
\[ \Rightarrow a + 7d = 22 . . . (ii)\]
\[\text { Solving (i) and (ii), we get: } \]
\[2d = 10\]
\[ \Rightarrow d = 5\]
\[\text { Putting the value of d in (i), we get: } \]
\[ a + 5 \times 5 = 12\]
\[ \Rightarrow a = 12 - 25 = - 13\]
\[ \therefore a_2 = a + \left( 2 - 1 \right)d = a + d = - 13 + 5 = - 8\]
\[\text { Also,} a_n = a + (n - 1)d \]
\[ = - 13 + (n - 1)5\]
\[ = - 13 + 5 n - 5 \]
\[ = 5n - 18 \]
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