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Question
The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.
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Solution
\[\text { Given }: \]
\[ a_3 = 7, a_7 - 3 a_3 = 2\]
\[\text { We have: } \]
\[ a_3 = 7\]
\[ \Rightarrow a + \left( 3 - 1 \right)d = 7\]
\[ \Rightarrow a + 2d = 7 . . . (i) \]
\[\text { Also }, a_7 - 3 a_3 = 2\]
\[ \Rightarrow a_7 - 21 = 2 (\text { Given })\]
\[ \Rightarrow a + \left( 7 - 1 \right)d = 23\]
\[ \Rightarrow a + 6d = 23 . . . (ii)\]
\[\text { From (i) and (ii), we get: } \]
\[4d = 16\]
\[ \Rightarrow d = 4\]
\[\text { Putting the value in (i), we get }: \]
\[ a + 2(4) = 7\]
\[ \Rightarrow a = - 1\]
\[ \therefore S_{20} = \frac{20}{2}\left[ 2\left( - 1 \right) + \left( 20 - 1 \right)(4) \right]\]
\[ \Rightarrow S_{20} = 10\left[ - 2 + 76 \right]\]
\[ \Rightarrow S_{20} = 10\left[ 74 \right] = 740\]
\[ \therefore a = - 1, d = 4, S_{20} = 740\]
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