Question

If $t_5,t_{10}$ and $t_{25}$ are $5^{th},{10}^{th}$ and ${25}^{th}$ terms of an AP respectively, then the value of $\left|\begin{array}{ccc}{t}_5& t_{10}& t_{25}\\ 5& 10& 25\\ 1& 1& 1\end{array}\right|$ is

• A. $-40$
• B. $1$
• C. $-1$
• D. $0$
• E. $40$

Answer 1

Answer: (D)

$0$

Let $a$ and $d$ be the first and common difference of an AP.
Then, $t_5=a+4d$
$t_{10}=a+9d$
and $t_{25}=a+24d$
Now, Let $△=\left|\begin{array}{ccc}{t}_5& t_{10}& t_{25}\\ 5& 10& 25\\ 1& 1& 1\end{array}\right|$
$=\left|\begin{array}{ccc}a+4d& a+9d& a+24d\\ 5& 10& 25\\ 1& 1& 1\end{array}\right|$
On applying operation
$C_2\to C_2-C_1$ and $C_3\to C_3-C_1$, we get
$△=\left|\begin{array}{ccc}a+4d& 5d& 20d\\ 5& 5& 20\\ 1& 0& 0\end{array}\right|$
On expanding along $R_3$, we get
$△=100d-100d=0$.