Matrices And Determinants

Let \(\mathrm{A_1,A_2,A_3}\) be the three A.P. with the same common difference d and having their first terms as \(\mathrm{A,A+1,A+2}\), respectively. Let a, b, c be the \(\mathrm{7^{th},9^{th},17^{th}}\) terms of \(\mathrm{A_1,A_2,A_3}\), respective such that \(\left| {\matrix{ a & 7 & 1 \cr {2b} & {17} & 1 \cr c & {17} & 1 \cr } } \right| + 70 = 0\).

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 ___________.