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  4. Let A1, A2, A3 be the three A . P. with the same common difference d and having their first terms as A , A +1, A +2, respectively, Let a, b, c be the 7 text th , 9 text th , 17 text th terms of A1, A2, A3, respectively such that |a 7 1 2 b 17 1 c 17 1|+70=0. If a=29, then the sum of first 20 terms of an AP whose first term is c-a-b and common difference is (d/12), is equal to

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Q. Let $A_1, A_2, A_3$ be the three $A . P$. with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively, Let $a, b, c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\begin{vmatrix}a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{vmatrix}+70=0$. If $a=29$, then the sum of first 20 terms of an AP whose first term is $c-a-b$ and common difference is $\frac{d}{12}$, is equal to

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Solution:

$ \begin{vmatrix} A +6 d & 7 & 1 \\ 2( A +1+8 d ) & 17 & 1 \\ A +2+16 d & 17 & 1\end{vmatrix}+70=0$
$ \Rightarrow A =-7 \text { and } d =6 $
$ \therefore c - a - b =20$
$ S _{20}=495$