Sequences & Series
AP with determinant condition
nta_pyq_2023_jan
Grade 11
Question:
Let $A_1, A_2, A_3$ be three A.P.s 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 $\begin{vmatrix} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1 \end{vmatrix} + 70 = 0$. If $a = 29$, then the sum of first 20 terms of an AP whose first term is $c - a - b$ and common difference is $\dfrac{d}{12}$, is equal to ______.
Step-by-Step Solution
Key Concept: Express $a, b, c$ in terms of $A$ and $d$: $a = A+6d$, $b = A+1+8d$, $c = A+2+16d$. Evaluate the determinant, set it equal to $-70$, and solve.
From determinant: $A = -7$, $d = 6$. So $c - a - b = 20$, $\frac{d}{12} = \frac{1}{2}$. $S_{20} = \frac{20}{2}[2(20) + 19 \cdot \frac{1}{2}] = 10[40 + \frac{19}{2}] = 10 \cdot \frac{99}{2} = 495$.
Correct Answer: 495