Question

If t$_5$, t$_{10}$ and t$_{25}$ are 5$^{th}$, 10$^{th}$, and 25$^{th}$ terms of an A.P. respectively, then the value of $\begin{vmatrix}t_{5}&t_{10}&t_{25}\\\ 5&10&25\\\ 1&1&1\end{vmatrix}$ is equal to

• A. -40
• B. 1
• C. -1
• D. 0

Answer 1

Answer: (D)

0

Answer 2

Let $a$ and $d$ be the first and common difference of an AP.
Then, $t_{5}=a+4 d$
$t_{10}=a+9 d$
and $t_{25}=a+24 d$
Now, Let $\Delta=\begin{vmatrix}t_{5} & t_{10} & t_{25} \\\ 5 & 10 & 25 \\\ 1 & 1 & 1\end{vmatrix}$
$=\begin{vmatrix}a+4 d & a+9 d & a+24 d \\\ 5 & 10 & 25 \\\ 1 & 1 & 1\end{vmatrix}$
On applying operation
$C _{2} \to C _{2}- C _{1}$ and $C _{3} \to C _{3}- C _{1}$, we get
$\Delta=\begin{vmatrix}a+4 d & 5 d & 20 d \\\5 & 5 & 20 \\\1 & 0 & 0\end{vmatrix}$
On expanding along $R_{3}$, we get
$\Delta=100 \,d-100\, d=0$