Question

If $a$, $b$, $c$ are respectively the $5^{\text{th}$, $8^{\text{th}}$, $13^{\text{th}}$ terms of an arithmetic progression, then}
$\begin{vmatrix} a & 5 & 1 \\ b & 8 & 1\\ c & 13 & 1 \end{vmatrix} = $

• A. $0$
• B. $1$
• C. $abc$
• D. $520$

Hint:

When values in a determinant follow a linear pattern, row operations often reveal simplifications like zero determinants.

Answer 1

The Correct Option is A

In an AP, we have: $a = a_1 + 4d$, $b = a_1 + 7d$, $c = a_1 + 12d$
Plugging into determinant:
$\begin{vmatrix} a_1 + 4d & 5 & 1 \\ a_1 + 7d & 8 & 1 \\ a_1 + 12d & 13 & 1 \end{vmatrix}$
Apply $R_1 \rightarrow R_1 - R_2$, $R_2 \rightarrow R_2 - R_3$ to make two rows identical $\Rightarrow$ determinant = 0